Question

1. $2x^2 + 16x - 32$

Explanation:

Step1: Factor out the GCF

Factor 2 from all terms.
$2(x^2 + 8x - 16)$

Step2: Check quadratic for factoring

Quadratic $x^2+8x-16$ has no integer factors, so we use the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ for $ax^2+bx+c=0$. Here $a=1$, $b=8$, $c=-16$.
First calculate discriminant: $\sqrt{8^2-4(1)(-16)}=\sqrt{64+64}=\sqrt{128}=8\sqrt{2}$

Step3: Find roots of quadratic

Substitute into quadratic formula.
$x=\frac{-8\pm8\sqrt{2}}{2}=-4\pm4\sqrt{2}$

Step4: Write fully factored form

Use roots to factor the quadratic inside the parentheses.
$2(x - (-4+4\sqrt{2}))(x - (-4-4\sqrt{2}))$

Answer:

$2(x + 4 - 4\sqrt{2})(x + 4 + 4\sqrt{2})$ (or factored GCF form: $2(x^2 + 8x - 16)$)