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