Question

select all the expressions that are equivalent to $(3x + 2)(x - 4)$ for all real values of $x$.
a $3x^2 - 12$
b $3x^2 - 10x - 8$
c $3(x^2 + 2x - 4)$
d $3(x^2 - 3x) - (x + 8)$
e $3x(x - 3) - 2(5x + 4)$
f $3x(x - 4) + 2(x - 4)$

Explanation:

Step1: Expand the original expression

First, use the distributive property (FOIL method) on $(3x+2)(x-4)$:
$$(3x)(x) + (3x)(-4) + 2(x) + 2(-4) = 3x^2 -12x +2x -8$$
Combine like terms:
$$3x^2 -10x -8$$

Step2: Simplify Option C

Distribute the 3 across the terms inside the parentheses:
$$3(x^2 +2x -4) = 3x^2 +6x -12$$

Step3: Simplify Option D

First distribute the 3, then subtract the second parentheses:
$$3(x^2 -3x) - (x+8) = 3x^2 -9x -x -8$$
Combine like terms:
$$3x^2 -10x -8$$

Step4: Simplify Option E

First distribute the $3x$ and $-2$, then combine terms:
$$3x(x-3) -2(5x+4) = 3x^2 -9x -10x -8$$
Combine like terms:
$$3x^2 -10x -8$$

Step5: Simplify Option F

First distribute the $3x$ and $2$, then combine terms:
$$3x(x-4) +2(x-4) = (3x+2)(x-4)$$
This is the original expression, which expands to:
$$3x^2 -10x -8$$

Answer:

B. $3x^2 -10x -8$
D. $3(x^2 - 3x) - (x + 8)$
E. $3x(x - 3) - 2(5x + 4)$
F. $3x(x - 4) + 2(x - 4)$