When you see a polynomial expression like this, you're being tested on factoring techniques. The goal is to break down the original expression into simpler factors that, when multiplied together, give you back the original polynomial.

Let's start by looking for a common factor in $$6x^3 + 9x^2 - 6x - 9$$. Each term is divisible by 3, so we can factor out 3: $$3(2x^3 + 3x^2 - 2x - 3)$$. Now we need to factor the expression inside the parentheses further.

For $$2x^3 + 3x^2 - 2x - 3$$, we can use grouping. Group the first two terms and the last two terms: $$x^2(2x + 3) - 1(2x + 3)$$. Notice that $$(2x + 3)$$ is a common factor, so we get $$(2x + 3)(x^2 - 1)$$. Since $$x^2 - 1$$ is a difference of squares, we could factor it further as $$(x + 1)(x - 1)$$, but none of our answer choices do this.

Therefore, our final answer is $$3(2x + 3)(x^2 - 1)$$, which matches choice D.

Choice A stops at the first step of factoring out 3. Choice B gives $$3(x + 1)(2x^2 - 3)$$, which expands to $$6x^3 + 6x^2 - 9x - 9$$ (incorrect middle terms). Choice C gives $$3(x - 1)(2x^2 + 5x + 3)$$, which expands to $$6x^3 + 9x^2 - 9x - 9$$ (wrong sign on the $$x$$ term).

Remember: always verify factoring by expanding your answer back to check if it matches the original expression.

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.

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.

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.

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.

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.

When you encounter a problem asking for the greatest common factor (GCF) of polynomial terms, you need to find the largest expression that divides evenly into all terms. This means examining both the numerical coefficients and the variable parts separately.

Let's break down each term in $$12x^3y^2 + 18x^2y^3 - 6xy^4$$:

• First term: $$12x^3y^2$$
• Second term: $$18x^2y^3$$
• Third term: $$6xy^4$$

For the coefficients (12, 18, -6), find their GCF by listing factors. The GCF of 12, 18, and 6 is 6.

For the variables, take the lowest power of each variable that appears in all terms:

• For $$x$$: the powers are 3, 2, and 1, so take $$x^1 = x$$
• For $$y$$: the powers are 2, 3, and 4, so take $$y^2$$

Therefore, the GCF is $$6xy^2$$, which is answer C.

Looking at the wrong answers: A) $$3xy$$ uses 3 instead of 6 for the coefficient—this fails to find the greatest numerical factor. B) $$6xy$$ correctly identifies the coefficient as 6 but uses $$y^1$$ instead of $$y^2$$—remember to take the lowest power that appears in all terms. D) $$6x^2y^2$$ incorrectly uses $$x^2$$ when the third term only has $$x^1$$, so $$x^2$$ doesn't divide evenly into all terms.

Strategy tip: Always check your GCF by dividing it into each original term—if any division leaves a remainder or fraction, your GCF is too large.

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.

When you see a quadratic expression like $$9x^2 - 30x + 25$$, you should immediately think about factoring, particularly checking if it's a perfect square trinomial. Perfect square trinomials follow the pattern $$a^2 - 2ab + b^2 = (a - b)^2$$ or $$a^2 + 2ab + b^2 = (a + b)^2$$.

To verify if this is a perfect square trinomial, identify what could be squared to give you the first and last terms. Since $$9x^2 = (3x)^2$$ and $$25 = 5^2$$, we have potential factors of $$3x$$ and $$5$$. Now check if the middle term fits: $$2 \cdot 3x \cdot 5 = 30x$$. Since our expression has $$-30x$$, this confirms we have $$(3x - 5)^2$$.

You can verify by expanding: $$(3x - 5)^2 = (3x)^2 - 2(3x)(5) + 5^2 = 9x^2 - 30x + 25$$ ✓

Choice A is correct. Choice B, $$(9x - 5)^2$$, would expand to $$81x^2 - 90x + 25$$, giving you the wrong coefficient for $$x^2$$. Choice C, $$(3x - 25)^2$$, would expand to $$9x^2 - 150x + 625$$, producing incorrect coefficients for both the middle and constant terms. Choice D, $$(3x + 5)^2$$, would expand to $$9x^2 + 30x + 25$$, giving you a positive middle term instead of negative.

Strategy tip: When factoring quadratics, always look for perfect square trinomials first by checking if the first and last terms are perfect squares, then verify the middle term equals twice their product.

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.