Example 11 Rationalize the denominator and simplify $\frac{2 \sqrt{6}}{\sqrt{3}+\sqrt{2}-\sqrt{5}}$.
$\begin{array}{l}\text { Solve the original expression }=\frac{(3+2 \sqrt{6}+2)-5}{\sqrt{3}+\sqrt{2}-\sqrt{5}} \\ =\frac{(\sqrt{3}+\sqrt{2})^{2}-(\sqrt{5})^{2}}{\sqrt{3}+\sqrt{2}-\sqrt{5}} \\ =\sqrt{3}+\sqrt{2}+\sqrt{5} .\end{array}$