Question

what is the completely factored form of the expression $16x^2 + 8x + 32$?
$4(4x^2 + 2x + 8)$
$4(12x^2 + 4x + 28)$
$8(2x^2 + x + 4)$
$8x(8x^2 + x + 24)$

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 factored form: $8(2x^2 + x + 4)$

Step3: Verify no further factoring

The quadratic $2x^2 + x + 4$ has discriminant $\Delta = 1^2 - 4(2)(4) = 1 - 32 = -31 < 0$, so it cannot be factored over real numbers.

Answer:

8(2x² + x + 4) (corresponding to the third option)