Answer
$ -6+3 \sqrt{5}+2 \sqrt{2}-\sqrt{10}$
Work Step by Step
Given \begin{equation}
\frac{3-\sqrt{2}}{2+\sqrt{5}}.
\end{equation} Rationalize the denominator and simplify.
\begin{equation}
\begin{aligned}
\frac{3-\sqrt{2}}{2+\sqrt{5}}&=\frac{(3-\sqrt{2})}{(2+\sqrt{5})}\cdot \frac{(2-\sqrt{5})}{(2-\sqrt{5})}\\
&= \frac{3(2-\sqrt{5})-\sqrt{2}(2-\sqrt{5}) }{4-5}\\
& = \frac{6-3 \sqrt{5}-2 \sqrt{2}+\sqrt{10}}{-1}\\
&= -6+3 \sqrt{5}+2 \sqrt{2}-\sqrt{10}.
\end{aligned}
\end{equation} The simplified expression is \begin{equation}
\frac{3-\sqrt{2}}{2+\sqrt{5}}= -6+3 \sqrt{5}+2 \sqrt{2}-\sqrt{10}.
\end{equation}
