Answer
The possible dimensions of the prism are $y,\ y+4$ and $3y+2$.
Work Step by Step
$3y^{3}+14y^{2}+8y=$
...factor out the GCF.
$=y(3y^{2}+14y+8)$
...find factors of $ac$ that have a sum of $b$.
($ac=24,\ b=14$)
$\left[\begin{array}{lll}
\text{Factors of 24 } & \text{Sum of factors} & \\
1\text{ and }12 & 13 & \\
2\text{ and }12 & 14 & \text{ is what we need}
\end{array}\right]$
...use the factors you found to rewrite $by$.
$=y(3y^{2}+12y+2y+8)$
...factor by grouping.
$=y[3y(y+4)+2(y+4)]$
...use the Distributive Property.
$=y(y+4)(3y+2)$
The possible dimensions of the prism are $y,\ y+4$ and $3y+2$.
