Answer
$\underline{{{x}^{2}}+2x-2}$
Work Step by Step
Consider
$f\left( x \right)={{x}^{3}}-6x+4$
Next, consider
$\begin{align}
& f\left( 2 \right)={{2}^{3}}-6\times 2+4 \\
& =0
\end{align}$
Therefore, using synthetic division, $x-2$ is a factor of the function $f\left( x \right)$ and hence
$\begin{align}
& f\left( x \right)={{x}^{3}}-6x+4 \\
& ={{x}^{3}}-2{{x}^{2}}+2{{x}^{2}}-4x-2x+4 \\
& ={{x}^{2}}\left( x-2 \right)+2x\left( x-2 \right)-2\left( x-2 \right) \\
& =\left( x-2 \right)\left( {{x}^{2}}+2x-2 \right)
\end{align}$
Therefore
$\frac{f\left( x \right)}{x-2}={{x}^{2}}+2x-2$
