Answer
a) $\frac{5m^4-44m^3-3m^2+308m-224}{m^2-7}=5m^2-44m+32$
b) $ (m^2-7)\cdot(m-8)\cdot(5m-4)$
Work Step by Step
Given \begin{equation}
\left(5m^4-44m^3-3m^2+308m-224\right) \div\left(m^2-7\right).
\end{equation} a) Use long division.
$$
\begin{array}{r}
5m^2-44m+32\phantom{)} \\
m^2-7{\overline{\smash{\big)}\,5m^4-44m^3-3m^2+308m-224\phantom{)}}}\\
\underline{-~\phantom{(}(5m^4-35m^2)\phantom{-b)}}\\
0-44m^3+32m^2+308m\phantom{)}\\
\underline{-~\phantom{()}(-44m^3+308m)}\\
0+32m^2-224\phantom{)}\\
\underline{-~\phantom{()}(32m^2-224)}\\
0\phantom{)}
\end{array}
$$ The solution is
\begin{equation}
\frac{5m^4-44m^3-3m^2+308m-224}{m^2-7}=5m^2-44m+32.
\end{equation} b) Factor the quadratic $5m^2-44m+32$.
\begin{equation}
\begin{aligned}
5m^2-44m+32&=5m^2-40m-4m+32 \\
&=5m(m-8)-4(m-8)\\
&= (m-8)(5m-4).
\end{aligned}
\end{equation} The prime factor decomposition is $$ (m^2-7)\cdot(m-8)\cdot(5m-4).$$
