\(\frac{\sqrt{3}+\sqrt{2}-1}{2+\sqrt{6}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+1}\cdot\left(\frac{\sqrt{3}}{2-\sqrt{6}}+\frac{\sqrt{3}}{2+\sqrt{6}}\right)-\frac{1}{\sqrt{2}}\)
\(=\frac{\sqrt{6}+2-\sqrt{2}-2-\sqrt{6}}{\sqrt{2}\left(2+\sqrt{6}\right)}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+1}\cdot\frac{2\sqrt{3}+3\sqrt{2}+2\sqrt{3}-3\sqrt{2}}{\left(2-\sqrt{6}\right)\left(2+\sqrt{6}\right)}\)
\(=\frac{-1}{2+\sqrt{6}}+\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+1}\cdot\frac{4\sqrt{3}}{-2}\)
\(=\frac{2-\sqrt{6}}{2}+\frac{6-2\sqrt{6}}{\sqrt{2}+1}\)
\(=\frac{\left(2-\sqrt{6}\right)\left(\sqrt{2}+1\right)+2\cdot\left(6-2\sqrt{6}\right)}{2\cdot\left(\sqrt{2}+1\right)}\)
\(=\frac{2\sqrt{2}+14-2\sqrt{3}-5\sqrt{6}}{2\cdot\left(\sqrt{2}+1\right)}\)
\(=\frac{\left(2\sqrt{2}+14-2\sqrt{3}-5\sqrt{6}\right)\left(\sqrt{2}-1\right)}{2}\)
\(=-5+6\sqrt{2}-4\sqrt{3}+\frac{3\sqrt{6}}{2}\)