Question

what is the completely factored form of the expression $16x^2 + 8x + 32?

$$8x\\left(8x^2 + x + 24\ ight)$$

8\left(2x^2 + x + 4\
ight)

$$4\\left(4x^2 + 2x + 8\ ight)$$

4\left(12x^2 + 4x + 28\
ight)$

Explanation:

Step1: Find GCF of coefficients

The coefficients are 16, 8, 32. Their greatest common factor (GCF) is 8.

Step2: Factor out the GCF

Divide each term by 8:
$16x^2 \div 8 = 2x^2$, $8x \div 8 = x$, $32 \div 8 = 4$
So $16x^2 + 8x + 32 = 8(2x^2 + x + 4)$

Step3: Verify inner trinomial

The trinomial $2x^2 + x + 4$ has no real factors (discriminant $\Delta = 1^2 - 4\times2\times4 = 1 - 32 = -31 < 0$), so it is fully factored.

Answer:

B. $8(2x^2 + x + 4)$