jmc

We have $f(2)=2^2+2\sqrt{2}=4+2\sqrt{2}$ so $2f(2)=8+4\sqrt{2}$. We also have $f(8)=8^2+2\sqrt{8}=64+2\cdot 2\sqrt{2}=64+4\sqrt{2}$. We subtract $8+4\sqrt{2}-(64+4\sqrt{2})=8+4\sqrt{2}-64-4\sqrt{2}=\boxed{-56}$.