Answer
a. $(2h^{3}+14h^2-72)\div(h-2)=2h^2+18h+36$
b. $h=2, l=9, w=4$
Work Step by Step
a.$\begin{array}{lllll}
\underline{2}| & 2 & 14 & 0 & -72\\
& & 4 & 36 & 72\\
\hline & & & & \\
& 2 & 18 & 36 & 0
\end{array}$
Since the last number on the bottom right of synthetic division known as the remainder is $0$, the equation divided perfectly. Therefore, $h=2$ is a solution to the equation.
$(2h^{3}+14h^2-72)\div(h-2)=2h^2+18h+36$
b. For a rectangular prism, the volume formula is, $V=h\times l \times w$
whereas, $h$ is height, $l$ is length and $w$ is width.
In our case, $h=h$, $l=h+7$ and $w=2h$, therefore the volume is, $V=2h^3+14h^2=72$,
$V=2h^3+14h^2-72=0$. The equation for Volume when the volume equals $72$ is the same as equation from part(a).
$V=2(h-2)(h^2+9h+18)=2(h-2)(h+3)(h+6)$
Therefore $h=2$ is the only positive solution to the equation.
$h=2, l=9, w=4$
