Factoring Expressions - ISEE Upper Level Mathematics Achievement

1. A walkway's area is $3x^2+12x$. What is the first step in factoring $3x^2+12x$?
   Answer: Factor out the GCF $3x$ This question tests ISEE Upper Level Mathematics Achievement skills, specifically identifying the first step in factoring expressions. Factoring involves breaking down an expression into a product of simpler expressions, and the first step should always be to look for a greatest common factor (GCF). For the expression 3x² + 12x, both terms share common factors: each term is divisible by 3 and by x, making the GCF equal to 3x. Choice A, "Factor out the GCF 3x," is correct because this is the standard first step in any factoring problem - we extract 3x to get 3x(x + 4). Choice B is incorrect because difference of squares doesn't apply to this expression (it's not in the form a² - b²), demonstrating a common error where students try to apply special factoring patterns before checking for common factors. Teaching strategies include creating a systematic factoring checklist with GCF as step one, practicing identification of common factors in each term, and emphasizing that factoring out the GCF often makes subsequent factoring steps easier and more obvious.
2. What is the complete factorization of $8x^3 - 27$?
   Answer: $(2x - 3)(4x^2 + 6x + 9)$ When you see an expression like $8x^3 - 27$, you should recognize this as a difference of cubes pattern: $a^3 - b^3$. The key insight is identifying what's being cubed: $8x^3 = (2x)^3$ and $27 = 3^3$. The difference of cubes formula is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. With $a = 2x$ and $b = 3$, we get: $8x^3 - 27 = (2x)^3 - 3^3 = (2x - 3)((2x)^2 + (2x)(3) + 3^2)$ Simplifying the second factor: $(2x)^2 + (2x)(3) + 3^2 = 4x^2 + 6x + 9$ Therefore: $8x^3 - 27 = (2x - 3)(4x^2 + 6x + 9)$ Choice A matches this exactly and is correct. Choice B has $(2x + 3)$ as the first factor, which would be wrong because we have subtraction, not addition, in our original expression. This represents the sum of cubes pattern instead. Choice C has the correct first factor but shows $4x^2 - 6x + 9$ in the second factor. The middle term should be $+6x$, not $-6x$, based on the difference of cubes formula. Choice D shows $(2x - 3)^3$, which would equal $8x^3 - 36x^2 + 54x - 27$—a completely different expression with additional terms. Study tip: Memorize both factoring formulas: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ and $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Notice how the signs alternate in specific patterns.
3. Which expression represents the factored form of $3x^2 - 12x - 15$?
   Answer: $3(x - 5)(x + 1)$ When factoring quadratic expressions, you should always look for a greatest common factor (GCF) first. Here, each term in $3x^2 - 12x - 15$ is divisible by 3, so factor that out: $3(x^2 - 4x - 5)$. Now you need to factor the quadratic $x^2 - 4x - 5$. You're looking for two numbers that multiply to $-5$ and add to $-4$. Those numbers are $-5$ and $+1$, since $(-5)(1) = -5$ and $(-5) + 1 = -4$. This gives you $x^2 - 4x - 5 = (x - 5)(x + 1)$. Therefore, the complete factorization is $3(x - 5)(x + 1)$, which is choice B. Choice A stops at the first step—factoring out the GCF—but doesn't complete the factorization of the quadratic expression inside the parentheses. While $3(x^2 - 4x - 5)$ is mathematically equivalent to the original expression, it's not fully factored. Choice C has the wrong signs in the binomial factors. If you expand $3(x + 5)(x - 1)$, you get $3(x^2 + 4x - 5) = 3x^2 + 12x - 15$, which has a positive middle term instead of negative. Choice D doesn't factor out the GCF properly. While $(3x - 15)(x + 1)$ does expand to the original expression, it's not the most reduced factored form since $3x - 15$ can be further factored as $3(x - 5)$. Remember: always factor out the GCF first, then look for patterns in what remains. This approach will lead you to the most simplified factored form.
4. If $ax^2 + bx + c$ factors as $(4x - 3)(2x + 1)$, what is the value of $abc$?
   Answer: $48$ This question tests your ability to work backward from a factored form to find the coefficients of a quadratic expression, then use those coefficients to calculate a specific value. To find the values of $a$, $b$, and $c$, you need to expand the given factorization $(4x - 3)(2x + 1)$. Using FOIL: - First: $4x \cdot 2x = 8x^2$ - Outer: $4x \cdot 1 = 4x$ - Inner: $-3 \cdot 2x = -6x$ - Last: $-3 \cdot 1 = -3$ Combining like terms: $8x^2 + 4x - 6x - 3 = 8x^2 - 2x - 3$ Therefore, $a = 8$, $b = -2$, and $c = -3$, giving us $abc = 8 \cdot (-2) \cdot (-3) = 48$. The answer is D. Let's examine why the other choices are incorrect. Choice A ($-48$) likely comes from making a sign error during the multiplication—perhaps calculating $8 \cdot 2 \cdot 3$ but incorrectly applying a negative sign. Choice B ($-24$) might result from correctly finding $a = 8$ and $c = -3$ but miscalculating $b$ as $-1$ instead of $-2$, giving $8 \cdot (-1) \cdot (-3) = 24$, then incorrectly applying a negative sign. Choice C ($24$) could come from the same coefficient error as B but without the extra negative sign. When expanding factored quadratics, always double-check your arithmetic, especially with negative signs. The FOIL method is reliable, but carefully combine like terms and watch for sign errors when multiplying negative coefficients.
5. If $6x^2 + kx - 5 = (2x - 1)(3x + 5)$, what is the value of $k$?
   Answer: $k = 7$ When you see an equation where a quadratic expression equals a factored form, you're looking at a polynomial identity problem. The key insight is that both sides must be equivalent for all values of the variable, so you can expand the right side and compare coefficients. Let's expand the right side: $(2x - 1)(3x + 5)$. Using FOIL: First terms give $2x \cdot 3x = 6x^2$, Outer terms give $2x \cdot 5 = 10x$, Inner terms give $-1 \cdot 3x = -3x$, and Last terms give $-1 \cdot 5 = -5$. Combining like terms: $6x^2 + 10x - 3x - 5 = 6x^2 + 7x - 5$. Now we can match coefficients with the left side $6x^2 + kx - 5$. The $x^2$ coefficients match (both are 6), the constant terms match (both are -5), and the $x$ coefficients must match, so $k = 7$. Looking at the wrong answers: A) $k = 4$ would give you $6x^2 + 4x - 5$, which doesn't match our expansion. C) $k = 10$ might tempt you if you only looked at the "outer" term ($10x$) but forgot to subtract the "inner" term ($-3x$). D) $k = 13$ could result from incorrectly adding $10 + 3 = 13$ instead of $10 - 3 = 7$. The correct answer is B) $k = 7$. Strategy tip: When checking polynomial identities, always expand completely and combine like terms before comparing coefficients. Don't try to shortcut by looking at individual terms from FOIL.
6. If $x^2 + bx + 20$ factors as $(x + 4)(x + 5)$, what is the value of $b$?
   Answer: $b = 9$ When you see a factored quadratic expression, you can use the relationship between the factors and the original polynomial to find missing coefficients. The key is understanding how FOIL (First, Outer, Inner, Last) connects the factored form back to standard form. Let's expand $(x + 4)(x + 5)$ using FOIL: - First: $x \cdot x = x^2$ - Outer: $x \cdot 5 = 5x$ - Inner: $4 \cdot x = 4x$ - Last: $4 \cdot 5 = 20$ Combining these terms: $x^2 + 5x + 4x + 20 = x^2 + 9x + 20$ Since this must equal $x^2 + bx + 20$, we can see that $b = 9$. Looking at the wrong answers: Choice (A) gives $b = 1$, which would mean the middle term is just $x$—far too small compared to our sum of $4x + 5x$. Choice (C) suggests $b = 10$, which might tempt you if you mistakenly think $b$ equals the product of the constants in the factors, but that's actually what gives us the last term (20). Choice (D) claims $b = 20$, which is the constant term, not the coefficient of $x$. Remember this pattern: when you have $(x + m)(x + n)$, the coefficient of $x$ in the expanded form is always $m + n$. Here, $4 + 5 = 9$, so the answer is (B). This shortcut can save you time—you don't always need to fully expand when you're just looking for the middle coefficient.
7. If $2x^2 - 8x + k$ can be factored as $2(x - a)^2$ for some constants $a$ and $k$, what is the value of $k$?
   Answer: $k = 8$ When you see a quadratic expression that can be factored in a specific form, you're working with completing the square or perfect square trinomials. The key insight is that both expressions must be equivalent, so their expanded forms must match exactly. Let's expand the given factored form $2(x - a)^2$ and compare it to $2x^2 - 8x + k$. First, expand $(x - a)^2 = x^2 - 2ax + a^2$, so $2(x - a)^2 = 2x^2 - 4ax + 2a^2$. Now we can match coefficients between $2x^2 - 8x + k$ and $2x^2 - 4ax + 2a^2$: - The $x^2$ coefficients already match: $2 = 2$ ✓ - The $x$ coefficients: $-8 = -4a$, so $a = 2$ - The constant terms: $k = 2a^2 = 2(2)^2 = 2(4) = 8$ Therefore, $k = 8$, which is choice B. Let's check why the other answers are wrong. Choice A ($k = 4$) likely comes from incorrectly calculating $a^2$ instead of $2a^2$. Choice C ($k = 12$) might result from adding $a^2 + 2a = 4 + 8$. Choice D ($k = 16$) could come from using $4a^2$ instead of $2a^2$. Strategy tip: When matching factored forms to expanded forms, always expand the factored version completely and match coefficients term by term. This systematic approach prevents calculation errors and ensures you don't miss any factors.
8. A rectangular patio has area $x^2-9$. Which expression is equivalent when factored completely?
   Answer: $(x-3)(x+3)$ This question tests ISEE Upper Level Mathematics Achievement skills, specifically factoring difference of squares. Factoring involves breaking down an expression into a product of simpler expressions, which is essential for simplifying expressions, solving equations, and understanding polynomial functions. For the given expression x² - 9, we recognize this as a difference of squares pattern a² - b² = (a-b)(a+b), where a = x and b = 3. Choice B, (x-3)(x+3), is correct because it follows the difference of squares formula, and when expanded using FOIL gives x² + 3x - 3x - 9 = x² - 9. Choice A is incorrect because (x-3)(x-3) = x² - 6x + 9, representing a perfect square trinomial rather than a difference of squares, a common error where students confuse these two special factoring patterns. Teaching strategies include memorizing the difference of squares pattern, using area models to visualize why (a-b)(a+b) = a² - b², and practicing recognition of perfect squares like 1, 4, 9, 16, 25 to quickly identify when this pattern applies.
9. Which expression is the complete factorization of $16x^4 - 1$?
   Answer: $(4x^2 + 1)(2x + 1)(2x - 1)$ When you see an expression like $16x^4 - 1$, recognize this as a difference of squares pattern: $a^2 - b^2 = (a+b)(a-b)$. Here, $16x^4 = (4x^2)^2$ and $1 = 1^2$, so you can factor this as $(4x^2 + 1)(4x^2 - 1)$. However, you're not done yet! Notice that $4x^2 - 1$ is itself another difference of squares: $4x^2 = (2x)^2$ and $1 = 1^2$. This means $4x^2 - 1 = (2x + 1)(2x - 1)$. So the complete factorization is: $16x^4 - 1 = (4x^2 + 1)(4x^2 - 1) = (4x^2 + 1)(2x + 1)(2x - 1)$, which is choice C. Choice A stops at the first factorization and doesn't factor $4x^2 - 1$ further. While $16x^4 - 1 = (4x^2 + 1)(4x^2 - 1)$ is correct, it's not completely factored. Choice B incorrectly treats $16x^4$ as $(4x)^2$ instead of $(4x^2)^2$. If you expand $(4x + 1)(4x - 1)$, you get $16x^2 - 1$, not $16x^4 - 1$. Choice D has the wrong exponents. Expanding $(2x + 1)^2(2x - 1)^2$ gives you $[(2x + 1)(2x - 1)]^2 = (4x^2 - 1)^2$, which equals $16x^4 - 8x^2 + 1$, not our original expression. Always factor completely when possible. With difference of squares, check if either factor can be factored further using the same pattern.
10. Which of the following expressions cannot be factored using integer coefficients?
    Answer: $x^2 + 5x + 7$ When you encounter quadratic expressions like these, you're being tested on factoring—specifically whether a quadratic can be written as a product of two binomials with integer coefficients. For an expression $x^2 + bx + c$, you need two integers that multiply to give $c$ and add to give $b$. Let's check each expression systematically. For choice A, $x^2 + 5x + 6$, you need two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3, so this factors as $(x + 2)(x + 3)$. For choice B, $x^2 + x - 6$, you need two numbers that multiply to -6 and add to 1. Those are 3 and -2, giving $(x + 3)(x - 2)$. Choice D, $x^2 - x - 6$, requires two numbers that multiply to -6 and add to -1. Those are -3 and 2, so it factors as $(x - 3)(x + 2)$. However, choice C, $x^2 + 5x + 7$, requires two integers that multiply to 7 and add to 5. Since 7 is prime, its only integer factor pairs are 1×7 and (-1)×(-7). The pair 1 and 7 adds to 8, while -1 and -7 adds to -8. Neither gives us the required sum of 5, making this expression impossible to factor using integer coefficients. The key strategy: when checking if a quadratic factors with integer coefficients, list all factor pairs of the constant term and see if any pair sums to the middle coefficient. If none work, it can't be factored over the integers.
11. Which of the following expressions is equivalent to $6x^3 + 9x^2 - 6x - 9$?
    Answer: $3(2x + 3)(x^2 - 1)$ Factor out the greatest common factor, then factor by grouping. Every term is divisible by $3$: $6x^3 + 9x^2 - 6x - 9 = 3(2x^3 + 3x^2 - 2x - 3)$. Group the terms inside: $x^2(2x + 3) - 1(2x + 3) = (2x + 3)(x^2 - 1)$. So the expression equals $3(2x + 3)(x^2 - 1)$. The other expressions fail when expanded. $3(2x^3 + 3x^2 - 2x + 3)$ ends in $+9$, not $-9$, because the sign of the last term was changed. $3(x + 1)(2x^2 - 3)$ expands to $6x^3 + 6x^2 - 9x - 9$. $3(x + 1)(2x^2 + 5x - 3)$ expands to $6x^3 + 21x^2 + 6x - 9$. The equivalent expression is $3(2x + 3)(x^2 - 1)$.
12. What is the factored form of $x^2 + 7x + 12$?
    Answer: $(x + 3)(x + 4)$ When you see a quadratic expression like $x^2 + 7x + 12$, you're looking at a factoring problem. The goal is to find two binomials that multiply together to give you the original expression. To factor $x^2 + 7x + 12$, you need two numbers that multiply to give you the constant term (12) and add to give you the middle coefficient (7). Let's think systematically: what pairs of numbers multiply to 12? The possibilities are 1×12, 2×6, and 3×4. Now check which pair adds to 7: 1+12=13, 2+6=8, and 3+4=7. Perfect! The numbers 3 and 4 work. Since both numbers are positive (they add to positive 7 and multiply to positive 12), both signs in the factored form will be positive: $(x + 3)(x + 4)$. You can verify this by expanding: $(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$. This confirms answer choice A is correct. Looking at the wrong answers: Choice B has $(x - 3)(x - 4)$, which expands to $x^2 - 7x + 12$ — the wrong sign on the middle term. Choice C gives $(x + 2)(x + 6) = x^2 + 8x + 12$ — wrong middle coefficient. Choice D gives $(x + 1)(x + 12) = x^2 + 13x + 12$ — also wrong middle coefficient. Remember: when factoring $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$. The signs of these numbers determine the signs in your factors.
13. What is the factored form of $x^3 + 27$?
    Answer: $(x + 3)(x^2 - 3x + 9)$ When you encounter an expression like $x^3 + 27$, you're looking at a sum of cubes, which follows the special factoring pattern: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Recognizing this pattern is crucial because sum of cubes cannot be factored using simpler methods. First, identify the cube roots: $x^3 = (x)^3$ and $27 = 3^3$, so we have $a = x$ and $b = 3$. Applying the sum of cubes formula: $x^3 + 27 = (x + 3)(x^2 - x \cdot 3 + 3^2) = (x + 3)(x^2 - 3x + 9)$. You can verify this by expanding: $(x + 3)(x^2 - 3x + 9) = x^3 - 3x^2 + 9x + 3x^2 - 9x + 27 = x^3 + 27$ ✓ Choice A $(x + 3)(x^2 + 9)$ incorrectly omits the middle term $-3x$ from the quadratic factor. This is a common error when students forget the complete sum of cubes pattern. Choice C $(x - 3)(x^2 + 3x + 9)$ uses the difference of cubes pattern instead. This would factor $x^3 - 27$, not $x^3 + 27$. Choice D $(x + 3)^3$ represents a perfect cube, which would expand to $x^3 + 9x^2 + 27x + 27$—completely different from our original expression. Study tip: Memorize both cube factoring formulas: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. Notice the sign patterns carefully—they're easy to mix up under test pressure.
14. Which expression is equivalent to $4x^2 - 12x + 9$?
    Answer: $(2x - 3)^2$ When you see a quadratic expression like $4x^2 - 12x + 9$, you should immediately check if it's a perfect square trinomial. These expressions follow the pattern $a^2 - 2ab + b^2 = (a - b)^2$. To factor $4x^2 - 12x + 9$, first identify what could be squared to give you the first and last terms. Since $4x^2 = (2x)^2$ and $9 = 3^2$, you're looking at a potential pattern where $a = 2x$ and $b = 3$. Check if the middle term fits: $2ab = 2(2x)(3) = 12x$. Since our expression has $-12x$, this confirms we have $(2x - 3)^2$. You can verify by expanding: $(2x - 3)^2 = (2x)^2 - 2(2x)(3) + 3^2 = 4x^2 - 12x + 9$ ✓ Now let's check why the other answers are wrong: Choice A: $4(x - 3)^2 = 4(x^2 - 6x + 9) = 4x^2 - 24x + 36$. The coefficients don't match our original expression. Choice C: $(4x - 3)^2 = 16x^2 - 24x + 9$. The first term becomes $16x^2$, not $4x^2$. Choice D: $(2x - 6)^2 = 4x^2 - 24x + 36$. Both the middle and last terms are incorrect. The correct answer is B. Study tip: When factoring quadratics, always check your work by expanding back to the original form. Perfect square trinomials are common on the ISEE, so practice recognizing the pattern $a^2 \pm 2ab + b^2$.
15. If $x^2 + mx + n = (x + 5)(x - 2)$, what is the value of $m + n$?
    Answer: $-7$ This question tests your ability to expand binomial expressions and match coefficients. When you see an equation where a quadratic expression equals a factored form, you need to expand the right side and compare coefficients. Let's expand $(x + 5)(x - 2)$ using FOIL or the distributive property: $(x + 5)(x - 2) = x^2 - 2x + 5x - 10 = x^2 + 3x - 10$ Now we can match this to the left side $x^2 + mx + n$: - The coefficient of $x^2$ is 1 (matches) - The coefficient of $x$ is $m = 3$ - The constant term is $n = -10$ Therefore, $m + n = 3 + (-10) = -7$. Looking at the wrong answers: Choice A gives $-13$, which you might get if you incorrectly calculated $m \cdot n = 3 \times (-10) = -30$ instead of $m + n$, or made an arithmetic error. Choice C gives $3$, which is just the value of $m$ alone—a common mistake when students forget to include $n$. Choice D gives $7$, which you'd get if you calculated $m + n$ but made a sign error, treating $n$ as positive 10 instead of negative 10. Strategy tip: When expanding factored expressions, always double-check your signs, especially with subtraction. Write out each step clearly: $(x + a)(x + b) = x^2 + (a+b)x + ab$. The middle coefficient is the sum of the constants, and the last term is their product.
16. A banner's area is $4x^2-25$. Which of the following represents the factored form?
    Answer: $(2x-5)(2x+5)$ This question tests ISEE Upper Level Mathematics Achievement skills, specifically factoring difference of squares with coefficients. Factoring involves breaking down an expression into a product of simpler expressions, which is essential for simplifying expressions and understanding polynomial functions. For 4x² - 25, we recognize this as a difference of squares where 4x² = (2x)² and 25 = 5², following the pattern a² - b² = (a-b)(a+b). Choice A, (2x-5)(2x+5), is correct because it applies the difference of squares formula with a = 2x and b = 5, and when expanded gives 4x² + 10x - 10x - 25 = 4x² - 25. Choice C is incorrect because (2x-5)² = 4x² - 20x + 25, which is a perfect square trinomial, not a difference of squares, showing confusion between these two important factoring patterns. Teaching strategies include recognizing perfect squares of expressions like (2x)² = 4x², using the FOIL method to verify factorizations, and creating visual models that show why the middle terms cancel in difference of squares.
17. If $ax^2 + bx + c = (3x - 2)(2x + 1)$, what is the value of $a - b + c$?
    Answer: $5$ When you see an equation where a quadratic expression equals a factored form, you're being asked to expand the right side and match coefficients. This tests your ability to multiply binomials and understand how the coefficients $a$, $b$, and $c$ relate to the factored form. To find $a$, $b$, and $c$, expand $(3x - 2)(2x + 1)$ using FOIL: - First: $3x \cdot 2x = 6x^2$ - Outer: $3x \cdot 1 = 3x$ - Inner: $(-2) \cdot 2x = -4x$ - Last: $(-2) \cdot 1 = -2$ Combining like terms: $6x^2 + 3x - 4x - 2 = 6x^2 - x - 2$ Therefore, $a = 6$, $b = -1$, and $c = -2$. Now calculate $a - b + c = 6 - (-1) + (-2) = 6 + 1 - 2 = 5$. Choice A ($3$) might result from incorrectly identifying $a = 3$ by looking only at the first coefficient in the factored form. Choice C ($7$) could come from sign errors when combining terms or miscalculating $6 - (-1) - (-2)$. Choice D ($9$) might occur if you mistakenly calculated $a + b + c$ instead of $a - b + c$, getting $6 + (-1) + (-2) = 3$, though even that doesn't equal $9$. The correct answer is B. Study tip: When expanding factored quadratics, always double-check your signs, especially with subtraction. Also, remember that $a - b + c$ is actually the value of the quadratic when $x = -1$, which can serve as a quick verification method.
18. Which of the following expressions is completely factored?
    Answer: $2x(x + 3)(x - 3)$ When you see "completely factored," you need to check whether each expression has been broken down into its simplest possible factors—meaning no factor can be factored further. Let's examine each option systematically. In choice C, $2x(x + 3)(x - 3)$, we have three factors: $2x$, $(x + 3)$, and $(x - 3)$. The factor $2x$ contains the prime number 2 and the variable $x$, neither of which can be factored further. The linear factors $(x + 3)$ and $(x - 3)$ are also in their simplest form. This expression is completely factored. Choice A, $x(x^2 - 4)$, looks factored but isn't complete. The factor $(x^2 - 4)$ is a difference of squares that can be factored further as $(x + 2)(x - 2)$. So this should be $x(x + 2)(x - 2)$. Choice B, $3(x^2 + 2x - 8)$, has the constant 3 factored out, but the quadratic $x^2 + 2x - 8$ can be factored as $(x + 4)(x - 2)$. The complete factorization would be $3(x + 4)(x - 2)$. Choice D, $(x^2 + 1)(x^2 - 9)$, contains $(x^2 - 9)$, which is another difference of squares that factors as $(x + 3)(x - 3)$. The factor $(x^2 + 1)$ cannot be factored further over the real numbers. Study tip: Always check for difference of squares patterns ($a^2 - b^2$) and factorable quadratics when verifying complete factorization. If you can factor any piece further, it's not completely factored.
19. If $2x^2 - 7x - 4 = (2x + 1)(x + k)$ for some constant $k$, what is the value of $k$?
    Answer: $k = -4$ When you see an equation where a quadratic expression equals a factored form, you're working with polynomial factorization. The key is to expand the right side and match coefficients with the left side. Let's expand $(2x + 1)(x + k)$ using the distributive property: $(2x + 1)(x + k) = 2x \cdot x + 2x \cdot k + 1 \cdot x + 1 \cdot k = 2x^2 + 2kx + x + k = 2x^2 + (2k + 1)x + k$ Now we can match this with the original expression $2x^2 - 7x - 4$: - The $x^2$ coefficients match: $2 = 2$ ✓ - The $x$ coefficients must match: $2k + 1 = -7$ - The constant terms must match: $k = -4$ From the $x$ coefficient equation: $2k + 1 = -7$, so $2k = -8$, which gives us $k = -4$. We can verify this with the constant term: $k = -4$ matches perfectly. Let's check why the other answers fail. Choice B gives $k = -2$, which would make the $x$ coefficient $2(-2) + 1 = -3$, not $-7$. Choice C gives $k = 2$, making the $x$ coefficient $2(2) + 1 = 5$, not $-7$. Choice D gives $k = 4$, making the $x$ coefficient $2(4) + 1 = 9$, not $-7$. Therefore, A is correct. Study tip: When factoring quadratics, always expand your answer to verify it matches the original expression. This double-check catches arithmetic errors and confirms your factorization is correct.
20. If $3x^2 + bx + 12$ factors as $(3x + 4)(x + 3)$, what is the value of $b$?
    Answer: $b = 13$ When you see a factoring problem where you're given the factored form and need to find a coefficient, you should expand the factored expression and compare it to the original polynomial. Let's expand $(3x + 4)(x + 3)$ using FOIL or the distributive property: $(3x + 4)(x + 3) = 3x \cdot x + 3x \cdot 3 + 4 \cdot x + 4 \cdot 3$ $= 3x^2 + 9x + 4x + 12$ $= 3x^2 + 13x + 12$ Since this must equal $3x^2 + bx + 12$, we can see that $b = 13$, making C the correct answer. Let's examine why the other choices are wrong. Choice A ($b = 9$) might come from only considering the first cross-multiplication term ($3x \cdot 3 = 9x$) and forgetting the second term. Choice B ($b = 11$) doesn't correspond to any natural step in the expansion process—it's likely a distractor. Choice D ($b = 15$) might result from incorrectly multiplying the constant terms of the binomials ($4 \times 3 = 12$, but somehow getting confused and using $4 + 3 + 4 + 3 + 1 = 15$ or another computational error). Remember: when working with factored polynomials, always expand completely and collect like terms. The middle term in a quadratic comes from adding the two cross-multiplication terms when you FOIL. Double-check your arithmetic, especially when combining like terms—this is where most errors occur in factoring problems.

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