\[\frac{1}{2\sqrt{2} + 2\sqrt{3} - \sqrt{6} - 2} =\]
\[= \frac{1}{2 \cdot \left( \sqrt{2} + \sqrt{3} \right) - \sqrt{2}\left( \sqrt{2} + \sqrt{3} \right)} =\]
\[= \frac{1}{\left( 2 - \sqrt{2} \right)\left( \sqrt{2} + \sqrt{3} \right)} =\]
\[= \frac{\left( 2 + \sqrt{2} \right)\left( \sqrt{2} - \sqrt{3} \right)}{(4 - 2)(2 - 3)} =\]
\[= \frac{\left( 2 + \sqrt{2} \right)\left( \sqrt{2} - \sqrt{3} \right)}{2 \cdot ( - 1)} =\]
\[= \frac{\left( 2 + \sqrt{2} \right)\left( \sqrt{3} - \sqrt{2} \right)}{2}\]