Question

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

Explanation:

Step1: Group terms by pairs

Group the first two and last two terms:
$(3x^3 - 12x^2) + (8x - 32)$

Step2: Factor out GCF from each group

Factor $3x^2$ from first group, 8 from second:
$3x^2(x - 4) + 8(x - 4)$

Step3: Factor out common binomial

Factor $(x - 4)$ from both terms:
$(3x^2 + 8)(x - 4)$
Verify by expanding option D to cross-check:
$(3x^2 + 4)(x - 4) = 3x^3 -12x^2 +4x -16$ (incorrect, correction: original grouping gives $(3x^2+8)(x-4)$, but matching to options, recheck factoring:
Wait, re-express original: $3x^2(x-4) +8(x-4) = (3x^2+8)(x-4)$. However, if we test option D expansion:
$(3x^2+4)(x-4)=3x^3-12x^2+4x-16$ (not matching). Recheck problem: original polynomial is $3x^3-12x^2+8x-32$. Correct factoring:
$3x^2(x-4) +8(x-4) = (3x^2+8)(x-4)$. But since this is not an option, re-verify grouping:
Alternative grouping: $3x^3+8x -12x^2-32 = x(3x^2+8) -4(3x^2+8) = (x-4)(3x^2+8)$.
Wait, perhaps a typo in options, but if we test each option:
Option A: $(3x^2-4)(x-4)=3x^3-12x^2-4x+16$ (no)
Option B: $(3x^2+4)(x+4)=3x^3+12x^2+4x+16$ (no)
Option C: $(3x^2-4)(x+4)=3x^3+12x^2-4x-16$ (no)
Option D: $(3x^2+4)(x-4)=3x^3-12x^2+4x-16$ (no)
Wait, correction: original polynomial $3x^3-12x^2+8x-32 = 3x^2(x-4) +8(x-4) = (3x^2+8)(x-4)$. But since this is not listed, recheck the problem: if the polynomial was $3x^3-12x^2+4x-16$, then D is correct. But given the problem as written, the correct factoring is $(3x^2+8)(x-4)$, but among the options, the closest is D, assuming a typo in the problem (8x should be 4x). Alternatively, recheck:
Wait, $3x^3-12x^2+8x-32 = 3x^2(x-4) +8(x-4) = (3x^2+8)(x-4)$. Since this is not an option, but if we factor by rational root theorem:
Possible roots: $\pm1,\pm2,\pm4,\pm8,\pm16,\pm32,\pm\frac{1}{3},\pm\frac{2}{3},\pm\frac{4}{3},\pm\frac{8}{3},\pm\frac{16}{3},\pm\frac{32}{3}$
Test x=4: $3(64)-12(16)+8(4)-32=192-192+32-32=0$. So (x-4) is a factor.
Divide $3x^3-12x^2+8x-32$ by (x-4):
Using polynomial division:
$3x^3-12x^2+8x-32 = (x-4)(3x^2+8)$
Since this is not an option, but the only option with (x-4) is D, which must be the intended answer (assuming a typo in the polynomial, 8x→4x).

Answer:

$\boldsymbol{(3x^2 + 8) (x - 4)}$ (matching to the provided options, the intended answer is likely $\boldsymbol{D. (3x^2 + 4) (x - 4)}$ assuming a coefficient typo in the original polynomial)