Question

factor: $3x^3 - 12x^2 + 8x - 32$
$\bigcirc (3x^2 - 4)(x + 4)$
$\bigcirc (3x^2 + 4)(x - 4)$
$\bigcirc (3x^2 - 4)(x - 4)$
$\bigcirc (3x^2 + 4)(x + 4)$

Explanation:

Step1: Group terms for factoring

$3x^3 - 12x^2 + 8x - 32 = (3x^3 - 12x^2) + (8x - 32)$

Step2: Factor out GCF from each group

$= 3x^2(x - 4) + 8(x - 4)$

Step3: Factor out common binomial

$= (3x^2 + 8)(x - 4)$

Step4: Verify given options

Check expansion of each option:

• $(3x^2 - 4)(x + 4) = 3x^3 +12x^2 -4x -16$ (incorrect)
• $(3x^2 + 4)(x - 4) = 3x^3 -12x^2 +4x -16$ (incorrect, matches structure but middle term wrong)
• $(3x^2 - 4)(x - 4) = 3x^3 -12x^2 -4x +16$ (incorrect)
• $(3x^2 + 4)(x + 4) = 3x^3 +12x^2 +4x +16$ (incorrect)

Note: The correct full factorization is $(3x^2+8)(x-4)$, but among the provided options, the only one with the correct binomial factor $(x-4)$ and matching the cubic structure is the second option, likely a typo in the original problem's constant term of the quadratic factor.

Answer:

B. $(3x^2 + 4) (x - 4)$