Answer
$\frac{-8+4\sqrt{6}+2\sqrt{7}-\sqrt{42}}{2}$
Work Step by Step
Given \begin{equation}
\frac{4-\sqrt{7}}{2+\sqrt{6}}.
\end{equation} Simplify by rationalizing the denominator.
\begin{equation}
\begin{aligned}
\frac{4-\sqrt{7}}{2+\sqrt{6}}&=\frac{(4-\sqrt{7})}{(2+\sqrt{6})}\cdot \frac{(2-\sqrt{6})}{(2-\sqrt{6})}\\
& =\frac{4(2-\sqrt{6})-\sqrt{7}(2-\sqrt{6})}{4-6}\\
&=\frac{8-4\sqrt{6}-2\sqrt{7}+\sqrt{42}}{-2}\\
&=\frac{-8+4\sqrt{6}+2\sqrt{7}-\sqrt{42}}{2}.
\end{aligned}
\end{equation}
