Answer
The possible dimensions of the prism are $4x,\ 3x+2$ and $3x+1$.
Work Step by Step
$36x^{3}+36x^{2}+8x=$
...factor out the GCF.
$=4x(9x^{2}+9x+2)$
...find factors of $ac$ that have a sum of $b$.
($ac=18,\ b=9$)
$\left[\begin{array}{lll}
\text{Factors of 18 } & \text{Sum of factors} & \\
1\text{ and }18 & 19 & \\
2\text{ and }9 & 11 & \\
3\text{ and }6 & 9 & \text{ is what we need}
\end{array}\right]$
...use the factors you found to rewrite $bx$.
$=4x(9x^{2}+3x+6x+2)$
...factor by grouping.
$=4x[3x(3x+1)+2(3x+1)]$
...use the Distributive Property.
$=4x(3x+2)(3x+1)$
The possible dimensions of the prism are $4x,\ 3x+2$ and $3x+1$.
