Answer
a) $\frac{8 x^3-86 x^2+237 x-189}{x-7}=8x^2-30x+27$
b) $ (x-7)\cdot (2x-3)\cdot(4x-9)$
Work Step by Step
Given \begin{equation}
\begin{aligned}
\left(8 x^3-86 x^2+237 x-189\right) \div(x-7).
\end{aligned}
\end{equation} a) Use synthetic division.
\begin{equation}
\begin{array}{r|rrrr}
7&8 & -86 & 237& -189\\
& & 56 & -210& 189\\
\hline &8& -30 & 27 & 0
\end{array}
\end{equation} The solution is
\begin{equation}
\begin{aligned}
\frac{8 x^3-86 x^2+237 x-189}{x-7}=8x^2-30x+27.
\end{aligned}
\end{equation} b) Factor the quadratic $8x^2-30x+27$.
\begin{equation}
\begin{aligned}
8x^2-30x+27&=8x^2-18x-12x+27 \\
&=2x(4x-9)-3(4x-9)\\
&= (2x-3)(4x-9).
\end{aligned}
\end{equation} The prime factor decomposition is $$ (x-7)\cdot (2x-3)\cdot(4x-9).$$
