Answer
$\dfrac{7+\sqrt{10}}{3}$
Work Step by Step
Rationalizing the denominator of $
\dfrac{4\sqrt{5}+\sqrt{2}}{2\sqrt{5}-\sqrt{2}}
$ results to
\begin{array}{l}
\dfrac{4\sqrt{5}+\sqrt{2}}{2\sqrt{5}-\sqrt{2}}
\cdot
\dfrac{2\sqrt{5}+\sqrt{2}}{2\sqrt{5}+\sqrt{2}}
\\\\=
\dfrac{(4\sqrt{5})(2\sqrt{5})+(4\sqrt{5})(\sqrt{2})+(\sqrt{2})(2\sqrt{5})+(\sqrt{2})(\sqrt{2})}{(2\sqrt{5})^2-(\sqrt{2})^2}
\\\\=
\dfrac{8\cdot5+4\sqrt{10}+2\sqrt{10}+2}{20-2}
\\\\=
\dfrac{40+6\sqrt{10}+2}{18}
\\\\=
\dfrac{42+6\sqrt{10}}{18}
\\\\=
\dfrac{7+\sqrt{10}}{3}
.\end{array}
