Answer
The possible dimensions of the prism are $4c,\ c+8$ and $c+5$.
Work Step by Step
$4c^{3}+52c^{2}+160c$
...factor out the GCF.
$=4c(c^{2}+13c+40)$
...find factors of $ac$ that have a sum of $b$.
($ac=40,\ b=13$)
$\left[\begin{array}{lll}
\text{Factors of 40 } & \text{Sum of factors} & \\
1\text{ and }40 & 41 & \\
2\text{ and }20 & 22 & \\
4\text{ and }10 & 14 & \\
5\text{ and }8 & 13 & \text{ is what we need }
\end{array}\right]$
...use the factors you found to rewrite $bc$.
$=4c(c^{2}+5c+8c+40)$
...factor by grouping.
$=4c[c(c+5)+8(c+5)]$
...use the Distributive Property.
$=4c(c+8)(c+5)$
The possible dimensions of the prism are $4c,\ c+8$ and $c+5$.
