Answer
a) $\frac{2 x^3+x^2-63 x-90}{x+5}=2x^2-9x-18$
b) $ (x+5)\cdot (x-6)\cdot(2x+3)$
Work Step by Step
Given \begin{equation}
\begin{aligned}
\left(2 x^3+x^2-63 x-90\right) \div(x+5).
\end{aligned}
\end{equation}
Use synthetic division.
\begin{equation}
\begin{array}{r|rrrr}
-5&2 & 1 & -63& -90\\
& & -10 & 45& 90\\
\hline &2& -9 & -18 & 0
\end{array}
\end{equation} a) The solution is
\begin{equation}
\begin{aligned}
\frac{2 x^3+x^2-63 x-90}{x+5}=2x^2-9x-18.
\end{aligned}
\end{equation} b) Factor the quadratic $2x^2-9x-18$.
\begin{equation}
\begin{aligned}
2x^2-9x-18&=2x^2+3x-12x-18 \\
&=x(2x+3)-6(2x+3)\\
&= (x-6)(2x+3).
\end{aligned}
\end{equation} The prime factor decomposition is $$ (x+5)\cdot (x-6)\cdot(2x+3).$$
