Answer
$16+10\sqrt{10}$
Work Step by Step
Using $(a+b)(c+d)=ac+ad+bc+bd$, or the product of 2 binomials, and the properties of radicals, the given expression, $
(4\sqrt{5}-3\sqrt{2})(2\sqrt{5}+4\sqrt{2})
,$ is equivalent to
\begin{array}{l}\require{cancel}
(4\sqrt{5})(2\sqrt{5})+(4\sqrt{5})(4\sqrt{2})-(3\sqrt{2})(2\sqrt{5})-(3\sqrt{2})(4\sqrt{2})
\\\\=
8\sqrt{25}+16\sqrt{10}-6\sqrt{10}-12\sqrt{4}
\\\\=
8\cdot5+16\sqrt{10}-6\sqrt{10}-12\cdot2
\\\\=
40+16\sqrt{10}-6\sqrt{10}-24
\\\\=
(40-24)+(16\sqrt{10}-6\sqrt{10})
\\\\=
16+10\sqrt{10}
.\end{array}
