\(\left(\dfrac{2\sqrt{3}-\sqrt{6}}{\sqrt{8}-2}+1\right)\cdot\dfrac{1}{2+\sqrt{6}}\)
\(=\left(\dfrac{\sqrt{6}\left(\sqrt{2}-1\right)}{2\left(\sqrt{2}-1\right)}+1\right)\cdot\dfrac{1}{2+\sqrt{6}}\)
\(=\dfrac{\sqrt{6}+2}{2}\cdot\dfrac{\sqrt{6}+2}{1}\)
\(=\dfrac{10+4\sqrt{6}}{2}=5+2\sqrt{6}\)
\(=\left(\dfrac{\sqrt{2}\left(\sqrt{6}-\sqrt{3}\right)}{2\left(\sqrt{2}-1\right)}+1\right).\dfrac{\sqrt{6}-2}{\left(\sqrt{6}+2\right)\left(\sqrt{6}-2\right)}\)
\(=\left(\dfrac{\left(\sqrt{6}-\sqrt{3}\right)}{\sqrt{2}\left(\sqrt{2}-1\right)}+1\right).\dfrac{\sqrt{6}-2}{6-4}\)
\(=\left(\dfrac{\left(\sqrt{6}-\sqrt{3}\right)}{2-\sqrt{2}}+1\right).\dfrac{\sqrt{6}-2}{2}\)
\(=\left(\dfrac{\left(\sqrt{6}-\sqrt{3}\right)\left(2+\sqrt{2}\right)}{\left(2-\sqrt{2}\right)\left(2+\sqrt{2}\right)}+1\right).\dfrac{\sqrt{6}-2}{2}\)
\(=\left(\dfrac{\sqrt{6}}{4-2}+1\right).\dfrac{\sqrt{6}-2}{2}\)
\(=\left(\dfrac{\sqrt{6}}{2}+1\right).\dfrac{\sqrt{6}-2}{2}\)
\(=\dfrac{2+\sqrt{6}}{2}.\dfrac{\sqrt{6}-2}{2}=\dfrac{2\sqrt{6}-4+6-2\sqrt{6}}{4}=\dfrac{1}{2}\)
