$\sqrt{2}(\sqrt{50}+\sqrt{8})=\sqrt{2}(\sqrt{25\ast 2}+\sqrt{4\ast 2})=\sqrt{2}(5\sqrt{2}+2\sqrt{2})=\sqrt{2}\ast 7\sqrt{2}=7\ast 2=14$

$(\sqrt{3}-\sqrt{12})\ast \sqrt{3}=(\sqrt{3}-\sqrt{4\ast 3})\ast \sqrt{3}=(\sqrt{3}-2\sqrt{3})\ast \sqrt{3}=-\sqrt{3}\ast \sqrt{3}=-3$

$(3\sqrt{5}-4\sqrt{3})\ast \sqrt{5}=3\sqrt{5}\ast \sqrt{5}-4\sqrt{3}\ast \sqrt{5}=3\ast 5-4\sqrt{3\ast 5}=15-4\sqrt{15}$

$2\sqrt{2}(3\sqrt{18}-\frac{1}{4}\sqrt{2}+\sqrt{32})=2\sqrt{2}(3\sqrt{9\ast 2}-\frac{1}{4}\sqrt{2}+\sqrt{16\ast 2})=2\sqrt{2}(3\ast 3\sqrt{2}-\frac{1}{4}\sqrt{2}+4\sqrt{2})=2\sqrt{2}(9\sqrt{2}-\frac{1}{4}\sqrt{2}+4\sqrt{2})=2\sqrt{2}\ast 12\frac{3}{4}\sqrt{2}=2\ast 2\ast \frac{51}{4}=4\ast \frac{51}{4}=51$