Question

1. which expression is equivalent to $-6x^2 - x + 12$?

a. $(3x + 2)(3 - 4x)$
b. $(3x - 2)(4x + 3)$
c. $- (2x - 3)(4 - 3x)$
d. $- (2x + 3)(3x - 4)$

1. select all the expressions that are equivalent to $3x^2 + 11x - 20$.

a. $(4x + 3)(5x - 1)$
b. $(4x - 3)(5x + 1)$
c. $(3x + 4)(x - 5)$
d. $(3x - 4)(x + 5)$

Explanation:

Question 8

Step 1: Expand option a

Expand \((3x + 2)(3 - 4x)\) using the distributive property (FOIL method):
\[

$$\begin{align*} (3x)(3)+(3x)(-4x)+(2)(3)+(2)(-4x)&=9x - 12x^{2}+6 - 8x\\ &=- 12x^{2}+x + 6 \end{align*}$$

\]
This is not equal to \(-6x^{2}-x + 12\).

Step 2: Expand option b

Expand \((3x - 2)(4x + 3)\):
\[

$$\begin{align*} (3x)(4x)+(3x)(3)+(-2)(4x)+(-2)(3)&=12x^{2}+9x-8x - 6\\ &=12x^{2}+x - 6 \end{align*}$$

\]
This is not equal to \(-6x^{2}-x + 12\).

Step 3: Expand option c

First, expand \((2x - 3)(4 - 3x)\):
\[

$$\begin{align*} (2x)(4)+(2x)(-3x)+(-3)(4)+(-3)(-3x)&=8x-6x^{2}-12 + 9x\\ &=-6x^{2}+17x - 12 \end{align*}$$

\]
Then multiply by \(-1\): \(-( - 6x^{2}+17x - 12)=6x^{2}-17x + 12\)
This is not equal to \(-6x^{2}-x + 12\).

Step 4: Expand option d

First, expand \((2x + 3)(3x - 4)\):
\[

$$\begin{align*} (2x)(3x)+(2x)(-4)+(3)(3x)+(3)(-4)&=6x^{2}-8x + 9x - 12\\ &=6x^{2}+x - 12 \end{align*}$$

\]
Then multiply by \(-1\): \(-(6x^{2}+x - 12)=-6x^{2}-x + 12\)
This matches the given expression.

Step 1: Expand option a

Expand \((4x + 3)(5x - 1)\):
\[

$$\begin{align*} (4x)(5x)+(4x)(-1)+(3)(5x)+(3)(-1)&=20x^{2}-4x + 15x - 3\\ &=20x^{2}+11x - 3 \end{align*}$$

\]
This is not equal to \(3x^{2}+11x - 20\).

Step 2: Expand option b

Expand \((4x - 3)(5x + 1)\):
\[

$$\begin{align*} (4x)(5x)+(4x)(1)+(-3)(5x)+(-3)(1)&=20x^{2}+4x-15x - 3\\ &=20x^{2}-11x - 3 \end{align*}$$

\]
This is not equal to \(3x^{2}+11x - 20\).

Step 3: Expand option c

Expand \((3x + 4)(x - 5)\):
\[

$$\begin{align*} (3x)(x)+(3x)(-5)+(4)(x)+(4)(-5)&=3x^{2}-15x + 4x - 20\\ &=3x^{2}-11x - 20 \end{align*}$$

\]
This is not equal to \(3x^{2}+11x - 20\).

Step 4: Expand option d

Expand \((3x - 4)(x + 5)\):
\[

$$\begin{align*} (3x)(x)+(3x)(5)+(-4)(x)+(-4)(5)&=3x^{2}+15x-4x - 20\\ &=3x^{2}+11x - 20 \end{align*}$$

\]
This matches the given expression.

Answer:

d. \(-(2x + 3)(3x - 4)\)

Question 9