smc

Multiply the numerator and the denominator by $\sqrt{2}-(\sqrt{3}+\sqrt{5})$ to get $\frac{2\sqrt{6}(\sqrt{2}-\sqrt{3}-\sqrt{5})}{(\sqrt{2}+(\sqrt{3}+\sqrt{5}))(\sqrt{2}-(\sqrt{3}+\sqrt{5})}$ $=\frac{2\sqrt{12}-2\sqrt{18}-2\sqrt{30}}{2-(\sqrt{3}+\sqrt{5}{)}^2}$ $=\frac{4\sqrt{3}-6\sqrt{2}-2\sqrt{30}}{-6-2\sqrt{15}}$ $=\frac{2\sqrt{3}-3\sqrt{2}-\sqrt{30}}{-3-\sqrt{15}}$. Now to get rid of the $\sqrt{15}$ in the denominator. Multiply the numerator and denominator by $-3+\sqrt{15}$ to get $\frac{(2\sqrt{3}-3\sqrt{2}-\sqrt{30})(-3+\sqrt{15})}{(-3-\sqrt{15})(-3+\sqrt{15})}$ $=\frac{-6\sqrt{3}+9\sqrt{2}+3\sqrt{30}+2\sqrt{45}-3\sqrt{30}-\sqrt{450}}{9-15}$ $=\frac{-6\sqrt{3}+9\sqrt{2}+3\sqrt{30}+6\sqrt{5}-3\sqrt{30}-15\sqrt{2}}{-6}$ $=\sqrt{2}+\sqrt{3}-\sqrt{5},\boxed{\sqrt{2}+\sqrt{3}-\sqrt{5}}$.