Answer
$3+2\sqrt{5}$
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})(2+\sqrt{5})
,$ is equivalent to
\begin{array}{l}\require{cancel}
4(2)+4(\sqrt{5})-\sqrt{5}(2)-\sqrt{5}(\sqrt{5})
\\\\=
8+4\sqrt{5}-2\sqrt{5}-\sqrt{5(5)}
\\\\=
8+4\sqrt{5}-2\sqrt{5}-\sqrt{(5)^2}
\\\\=
8+4\sqrt{5}-2\sqrt{5}-5
\\\\=
(8-5)+(4\sqrt{5}-2\sqrt{5})
\\\\=
3+2\sqrt{5}
.\end{array}
