Answer
$(a+4b)(a+8b)$
Work Step by Step
$12$ is positive, so we list positive factors of $32$.
$\left[\begin{array}{lll}
\text{Factors of 32 } & \text{Sum of factors}1 \text{ and }32 & 33\\
2\text{ and }16 & 18 & \\
4\text{ and }8 & 12 & \text{...is what we need}
\end{array}\right]$
$a^{2}+12ab+32b^{2}=(a+4b)(a+8b)$
Check:
$(a+4b)(a+8b)=$
...use the FOIL method.
$=a^{2}+8ab+4ab+32b^{2}$
...add like terms.
$=a^{2}+12ab+32b^{2}$
