Question

select all the expressions that are equivalent to $(3x + 2)(x - 4)$ for all real values
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 target expression

$$(3x+2)(x-4) = 3x\cdot x + 3x\cdot(-4) + 2\cdot x + 2\cdot(-4) = 3x^2 -12x +2x -8 = 3x^2 -10x -8$$

Step2: Check Option A

$3x^2 -12
eq 3x^2 -10x -8$

Step3: Check Option B

$3x^2 -10x -8$ matches the expanded target.

Step4: Check Option C

$$3(x^2 +2x -4) = 3x^2 +6x -12 eq 3x^2 -10x -8$$

Step5: Check Option D

$$3(x^2 -3x)-(x+8) = 3x^2 -9x -x -8 = 3x^2 -10x -8$$, matches the target.

Step6: Check Option E

$$3x(x-3)-2(5x+4) = 3x^2 -9x -10x -8 = 3x^2 -19x -8 eq 3x^2 -10x -8$$

Step7: Check Option F

$$3x(x-4)+2(x-4) = (3x+2)(x-4)$$, which is the original target expression, so it is equivalent.

Answer:

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