Answer
$4+\sqrt[]{10}+4\sqrt{3}+\sqrt{30}$
Work Step by Step
Using $(a+b)(c+d)=ac+ad+bc+bd$, or the Distributive Property, and the properties of radicals, the given expression, $
(\sqrt[]{2}+\sqrt{6})(\sqrt[]{8}+\sqrt{5})
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt[]{2}(\sqrt[]{8})+\sqrt[]{2}(\sqrt{5})+\sqrt{6}(\sqrt[]{8})+\sqrt{6}(\sqrt{5})
\\\\=
\sqrt[]{16}+\sqrt[]{10}+\sqrt{48}+\sqrt{30}
\\\\=
\sqrt[]{(4)^2}+\sqrt[]{10}+\sqrt{16\cdot3}+\sqrt{30}
\\\\=
\sqrt[]{(4)^2}+\sqrt[]{10}+\sqrt{(4)^2\cdot3}+\sqrt{30}
\\\\=
4+\sqrt[]{10}+4\sqrt{3}+\sqrt{30}
.\end{array}