Answer
$5 x^2+7x+3$.
Work Step by Step
The area of the rectangle is $$A= 5 x^3+27 x^2+31 x+12$$ and the length of the shortest side is $$w= x+4.$$We know that the area of a rectangle is $$A= l\times w.$$. The length of the missing side is given by:
\begin{equation}
\begin{aligned}
l&=\frac{A}{w}\\
&= \frac{5 x^3+27 x^2+31 x+12}{x+4}.
\end{aligned}
\end{equation} Use synthetic division.
\begin{equation}
\begin{array}{r|rrrr}
-4& 5 & 27 & 31 & 12 \\
& & -20 & -28 & -12\\
\hline & 5 & 7 & 3&0
\end{array}
\end{equation} The missing length is $5 x^2+7x+3$.
