Factor by recognizing it as a perfect square trinomial.
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Algebra Quiz
Practice Using Structure To Rewrite Expressions in Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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Factor x2−10x+25 by recognizing it as a perfect square trinomial.
This quiz focuses on Using Structure To Rewrite Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra.
Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.
Factor x2−10x+25 by recognizing it as a perfect square trinomial.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! For x² - 10x + 25, x² = (x)², 25 = 5², and -10x = -2 · x · 5, fitting (x - 5)² perfectly. Choice A correctly recognizes the pattern as a perfect square trinomial and rewrites it as (x - 5)². Excellent pattern recognition! Choice B has the wrong sign for the middle term—remember to match the signs in the pattern; checking by expanding will help solidify this! For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
Rewrite 9x2−24x+16 using its structure as a perfect square trinomial.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! In 9x² - 24x + 16, 9x² = (3x)², 16 = 4², and -24x = -2 · 3x · 4, matching the negative pattern for (3x - 4)². Choice A correctly recognizes the pattern as a perfect square trinomial and rewrites it as (3x - 4)². Excellent pattern recognition! Choice B flips the sign, but verify by checking the middle term—it should be negative here; expanding is a helpful double-check! For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
See x4−1 as (x2)2−12 and factor completely over the integers.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in x4−y4, if we see it as (x2)2−(y2)2, we recognize it's a difference of squares and can factor it as (x2+y2)(x2−y2), then factor the second part again since it's also a difference of squares! Here, x4−1 is (x2)2−12=(x2−1)(x2+1), and x2−1 factors further to (x−1)(x+1). Choice B correctly factors completely to (x−1)(x+1)(x2+1). Excellent pattern recognition! Choice A doesn't factor all the way—always check for more patterns in the factors; it's a skill that gets easier with practice! When you see higher powers like x4, think: can I view this as a perfect square? x4=(x2)2, so x4−1 becomes (x2)2−12= difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For x4−16=(x2+4)(x2−4), the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
What is x4−16 when factored completely? (Hint: view it as (x2)2−42.)
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in x⁴ - y⁴, if we see it as (x²)² - (y²)², we recognize it's a difference of squares and can factor it as (x² + y²)(x² - y²), then factor the second part again since it's also a difference of squares! Here, x⁴ - 16 is (x²)² - 4², factoring to (x² - 4)(x² + 4), and then x² - 4 is another difference of squares: (x - 2)(x + 2). Choice B correctly factors completely to (x - 2)(x + 2)(x² + 4). Excellent pattern recognition! Choice D might come from thinking it's a perfect square, but expand it to check—it gives extra terms; keep practicing by verifying your factors multiply back correctly! When you see higher powers like x⁴, think: can I view this as a perfect square? x⁴ = (x²)², so x⁴ - 1 becomes (x²)² - 1² = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For x⁴ - 16 = (x² + 4)(x² - 4), the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
Recognize x2−12x+36 as what type of pattern, and rewrite it in factored form.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! Let's check if x² - 12x + 36 fits a perfect square pattern. First term: x² is a perfect square. Last term: 36 = 6², also a perfect square. Middle term: -12x should equal -2·x·6 = -12x. Perfect! Since we have positive-negative-positive signs, this matches the (a - b)² pattern where a = x and b = 6. Therefore, x² - 12x + 36 = (x - 6)². Choice A correctly shows this as (x - 6)². Excellent pattern recognition! Choice D would give us (x - 6)(x + 6) = x² - 36, which is missing the middle term entirely—that's a difference of squares, not a perfect square trinomial. For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
Recognize x2+12x+36 as a perfect square trinomial and rewrite it as a squared binomial.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! For x² + 12x + 36, notice x² is (x)², 36 is 6², and 12x = 2 · x · 6, with all positive signs, so it matches (x + 6)². Choice A correctly recognizes the pattern as a perfect square trinomial and rewrites it as (x + 6)². Excellent pattern recognition! Choice B might tempt if you mix up the signs, but check the middle term's sign—it should match the pattern for addition here; don't worry, verifying by expanding helps build that intuition! For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
See x4−81 as a difference of squares by viewing it as (x2)2−92, and factor completely.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. The difference of squares pattern a² - b² = (a + b)(a - b) is super useful: whenever you see two perfect squares being subtracted (with no middle term), you can factor it as the sum and difference of what's being squared. For example, x² - 9 = (x)² - (3)² = (x + 3)(x - 3). Here we have x⁴ - 81, which we can view as (x²)² - 9². Applying the difference of squares formula: (x²)² - 9² = (x² + 9)(x² - 9). But wait—we're not done! The factor (x² - 9) is itself a difference of squares: x² - 9 = (x + 3)(x - 3). So the complete factorization is (x² + 9)(x + 3)(x - 3). Choice C correctly shows this complete factorization with all three factors. Excellent pattern recognition! Choice A stops too early—it factors the first difference of squares but doesn't recognize that x² - 9 can be factored further. When you see higher powers like x⁴, think: can I view this as a perfect square? x⁴ = (x²)², so x⁴ - 1 becomes (x²)² - 1² = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For x⁴ - 16 = (x² + 4)(x² - 4), the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
See 16x2−9 as a difference of squares and rewrite it as a product of two binomials.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. The difference of squares pattern a2−b2=(a+b)(a−b) is super useful: whenever you see two perfect squares being subtracted (with no middle term), you can factor it as the sum and difference of what's being squared. For example, x2−9=(x)2−(3)2=(x+3)(x−3). Let's examine 16x2−9 for the difference of squares pattern. First term: 16x2=(4x)2, which is a perfect square. Second term: 9=32, also a perfect square. They're being subtracted with no middle term—we have a difference of squares! Applying the formula: (4x)2−32=(4x+3)(4x−3). Choice B correctly identifies this as (4x+3)(4x−3). Excellent pattern recognition! Choice A shows (16x−3)(x+3), but if you multiply this out, you get 16x2+48x−3x−9=16x2+45x−9, which has a middle term our original doesn't have. To spot difference of squares: (1) Are there exactly two terms? (2) Are they being subtracted? (3) Is each term a perfect square (like x2, 9, 4x2, 25)? If yes to all three, you've got a2−b2, which factors as (a+b)(a−b). Try it with x2−16: yes two terms, yes subtraction, yes both perfect squares → (x+4)(x−4). Boom!
See 16x4−9 as a difference of squares and factor completely.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in x⁴ - y⁴, if we see it as (x²)² - (y²)², we recognize it's a difference of squares and can factor it as (x² + y²)(x² - y²), then factor the second part again since it's also a difference of squares! Looking at 16x⁴ - 9, we need to recognize what's being squared: 16x⁴ = (4x²)² and 9 = 3². So we have (4x²)² - 3², which is a difference of squares with a = 4x² and b = 3. Applying the pattern (a² - b² = (a + b)(a - b)) gives us (4x² + 3)(4x² - 3). Choice A correctly recognizes 16x⁴ - 9 as (4x²)² - 3² and applies the difference of squares formula to get (4x² + 3)(4x² - 3). Excellent pattern recognition! Choice D would only work if we had 16x² - 9, not 16x⁴—watch those exponents carefully! When you see higher powers like x⁴, think: can I view this as a perfect square? x⁴ = (x²)², so x⁴ - 1 becomes (x²)² - 1² = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first.
See 16x4−81 as a difference of squares by viewing it as (4x2)2−92, and factor completely over the integers.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in x4−y4, if we see it as (x2)2−(y2)2, we recognize it's a difference of squares and can factor it as (x2+y2)(x2−y2), then factor the second part again since it's also a difference of squares! For 16x4−81, view it as (4x2)2−92, which factors to (4x2−9)(4x2+9), and then factor 4x2−9 further as (2x−3)(2x+3) for complete factorization. Choice C correctly applies the difference of squares twice to factor completely to (2x−3)(2x+3)(4x2+9). Excellent pattern recognition! Choice A stops too early without factoring further—remember to check if factors can be broken down more; it's a common step, and you're getting better at it! When you see higher powers like x4, think: can I view this as a perfect square? x4=(x2)2, so x4−1 becomes (x2)2−12 = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For x4−16=(x2+4)(x2−4), the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
Which shows 4x2−25 rewritten to reveal its structure as a difference of squares?
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. The difference of squares pattern a² - b² = (a + b)(a - b) is super useful: whenever you see two perfect squares being subtracted (with no middle term), you can factor it as the sum and difference of what's being squared. For example, x² - 9 = (x)² - (3)² = (x + 3)(x - 3). For 4x² - 25, it's (2x)² - 5², so it factors to (2x + 5)(2x - 5). Choice C correctly recognizes the pattern as difference of squares and applies the formula to get (2x + 5)(2x - 5). Excellent pattern recognition! Choice A might confuse it with a perfect square, but there's no middle term here—look for that subtraction between squares; you're improving with each try! To spot difference of squares: (1) Are there exactly two terms? (2) Are they being subtracted? (3) Is each term a perfect square (like x², 9, 4x², 25)? If yes to all three, you've got a² - b², which factors as (a + b)(a - b). Try it with x² - 16: yes two terms, yes subtraction, yes both perfect squares → (x + 4)(x - 4). Boom!
See x4−y4 as (x2)2−(y2)2 and factor completely.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in x⁴ - y⁴, if we see it as (x²)² - (y²)², we recognize it's a difference of squares and can factor it as (x² + y²)(x² - y²), then factor the second part again since it's also a difference of squares! So for x⁴ - y⁴, first (x² + y²)(x² - y²), then factor x² - y² to (x + y)(x - y), giving (x + y)(x - y)(x² + y²) completely. Choice B correctly applies the formula and factors completely to get (x - y)(x + y)(x² + y²). Excellent pattern recognition! If you stopped at A, good start, but check for further factoring in each binomial. When you see higher powers like x⁴, think: can I view this as a perfect square? x⁴ = (x²)², so x⁴ - 1 becomes (x²)² - 1² = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For x⁴ - 16 = (x² + 4)(x² - 4), the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
See x4−1 as (x2)2−12 and factor completely over the integers.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in x⁴ - y⁴, if we see it as (x²)² - (y²)², we recognize it's a difference of squares and can factor it as (x² + y²)(x² - y²), then factor the second part again since it's also a difference of squares! Here, x⁴ - 1 is (x²)² - 1² = (x² - 1)(x² + 1), and x² - 1 factors further to (x - 1)(x + 1). Choice B correctly factors completely to (x - 1)(x + 1)(x² + 1). Excellent pattern recognition! Choice A doesn't factor all the way—always check for more patterns in the factors; it's a skill that gets easier with practice! When you see higher powers like x⁴, think: can I view this as a perfect square? x⁴ = (x²)², so x⁴ - 1 becomes (x²)² - 1² = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For x⁴ - 16 = (x² + 4)(x² - 4), the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
Which shows 4x2−25 rewritten to reveal its structure as a difference of squares?
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. The difference of squares pattern a² - b² = (a + b)(a - b) is super useful: whenever you see two perfect squares being subtracted (with no middle term), you can factor it as the sum and difference of what's being squared. For example, x² - 9 = (x)² - (3)² = (x + 3)(x - 3). For 4x² - 25, it's (2x)² - 5², so it factors to (2x + 5)(2x - 5). Choice C correctly recognizes the pattern as difference of squares and applies the formula to get (2x + 5)(2x - 5). Excellent pattern recognition! Choice A might confuse it with a perfect square, but there's no middle term here—look for that subtraction between squares; you're improving with each try! To spot difference of squares: (1) Are there exactly two terms? (2) Are they being subtracted? (3) Is each term a perfect square (like x², 9, 4x², 25)? If yes to all three, you've got a² - b², which factors as (a + b)(a - b). Try it with x² - 16: yes two terms, yes subtraction, yes both perfect squares → (x + 4)(x - 4). Boom!
Use structure to factor by grouping: x3+3x2+2x+6.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Using structure means looking for patterns you already know—difference of squares, perfect square trinomials, common factors—rather than just trying random factor pairs. It's like having a toolkit of patterns that make factoring much faster when you spot them! For x³ + 3x² + 2x + 6, group as (x³ + 3x²) + (2x + 6) = x²(x + 3) + 2(x + 3), then factor out the common (x + 3) to get (x² + 2)(x + 3). Choice A correctly applies grouping to factor completely to (x + 3)(x² + 2). Excellent pattern recognition! If you selected B, the terms are close but the grouping doesn't match—double-check by expanding to verify. After factoring using a pattern, always check: (1) Can any factor be factored further? (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
Recognize x2+12x+36 as a perfect square trinomial and rewrite it as a squared binomial.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! For x² + 12x + 36, notice x² is (x)², 36 is 6², and 12x = 2 · x · 6, with all positive signs, so it matches (x + 6)². Choice A correctly recognizes the pattern as a perfect square trinomial and rewrites it as (x + 6)². Excellent pattern recognition! Choice B might tempt if you mix up the signs, but check the middle term's sign—it should match the pattern for addition here; don't worry, verifying by expanding helps build that intuition! For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
Factor 2x3+8x2+3x+12 by grouping (factor out the common binomial).
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Using structure means looking for patterns you already know—difference of squares, perfect square trinomials, common factors—rather than just trying random factor pairs. It's like having a toolkit of patterns that make factoring much faster when you spot them! For 2x³ + 8x² + 3x + 12, let's group the terms: (2x³ + 8x²) + (3x + 12). From the first group, factor out 2x²: 2x²(x + 4). From the second group, factor out 3: 3(x + 4). Now we have 2x²(x + 4) + 3(x + 4). Both terms share the common binomial factor (x + 4)! Factoring it out: (x + 4)(2x² + 3). Choice A correctly shows this as (x + 4)(2x² + 3). Excellent pattern recognition! Choice B shows (2x + 3)(x² + 4), which would expand to 2x³ + 8x + 3x² + 12 = 2x³ + 3x² + 8x + 12. The x² and x terms are in the wrong order compared to our original expression. When factoring by grouping: (1) Group terms that might share a common factor, (2) Factor out the GCF from each group, (3) Look for a common binomial factor, (4) Factor out that common binomial. Always verify by expanding your answer—it should match the original expression!
What is x4−16 when factored completely? (Hint: view it as (x2)2−42.)
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. Sometimes viewing parts of an expression as single entities helps reveal structure: in x⁴ - y⁴, if we see it as (x²)² - (y²)², we recognize it's a difference of squares and can factor it as (x² + y²)(x² - y²), then factor the second part again since it's also a difference of squares! Here, x⁴ - 16 is (x²)² - 4², factoring to (x² - 4)(x² + 4), and then x² - 4 is another difference of squares: (x - 2)(x + 2). Choice B correctly factors completely to (x - 2)(x + 2)(x² + 4). Excellent pattern recognition! Choice D might come from thinking it's a perfect square, but expand it to check—it gives extra terms; keep practicing by verifying your factors multiply back correctly! When you see higher powers like x⁴, think: can I view this as a perfect square? x⁴ = (x²)², so x⁴ - 1 becomes (x²)² - 1² = difference of squares! This 'strategic viewing' lets you use patterns you already know on expressions that look different at first. After factoring using a pattern, always check: (1) Can any factor be factored further? For x⁴ - 16 = (x² + 4)(x² - 4), the second factor is another difference of squares! (2) Do your factors multiply back to the original? FOIL or expand to verify. These checks catch mistakes and build confidence!
Recognize 9x2+30x+25 as what type of pattern and rewrite it accordingly.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. A perfect square trinomial has the form a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²: the first and last terms are perfect squares, and the middle term is exactly twice their product. Recognizing this pattern lets you write the trinomial as a single squared binomial! Let's check 9x² + 30x + 25: first term 9x² = (3x)², last term 25 = 5², and for the pattern a² + 2ab + b², the middle term should be 2 · 3x · 5 = 30x. Perfect match! This is a perfect square trinomial with a = 3x and b = 5, giving us (3x + 5)². Choice B correctly identifies this as a perfect square trinomial and rewrites it as (3x + 5)². Excellent pattern recognition! Choice A incorrectly identifies it as a difference of squares, but we have addition, not subtraction—and three terms, not two! For perfect square trinomials, check three things: (1) First and last terms are perfect squares, (2) Middle term has the right coefficient (twice the product of what you're squaring), (3) Signs match the pattern (++ means (a+b)², +- means (a-b)²). Example: x² + 10x + 25 has x² (✓), 25 = 5² (✓), and 10x = 2·x·5 (✓), so it's (x + 5)²!
Rewrite 4x2−25 using its structure as a difference of squares.
Explanation: This question tests your ability to recognize algebraic patterns and structures—like difference of squares or perfect square trinomials—that let you rewrite expressions more efficiently. The difference of squares pattern a² - b² = (a + b)(a - b) is super useful: whenever you see two perfect squares being subtracted (with no middle term), you can factor it as the sum and difference of what's being squared. For example, x² - 9 = (x)² - (3)² = (x + 3)(x - 3). Looking at 4x² - 25, we need to identify what's being squared: 4x² = (2x)² and 25 = 5². So we have (2x)² - 5², which is a difference of squares with a = 2x and b = 5. Applying the pattern gives us (2x + 5)(2x - 5). Choice B correctly recognizes the pattern as difference of squares and applies the formula (2x)² - 5² = (2x + 5)(2x - 5). Excellent pattern recognition! Choice A has the wrong coefficient on x in the first factor—remember that 4x² = (2x)², not (4x)². To spot difference of squares: (1) Are there exactly two terms? (2) Are they being subtracted? (3) Is each term a perfect square (like x², 9, 4x², 25)? If yes to all three, you've got a² - b², which factors as (a + b)(a - b). Try it with x² - 16: yes two terms, yes subtraction, yes both perfect squares → (x + 4)(x - 4). Boom!