Answer
$-18-\sqrt{5}$
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, $
(3+2\sqrt{5})(4-3\sqrt{5})
,$ simplifies to
\begin{array}{l}\require{cancel}
3(4)+3(-3\sqrt{5})+2\sqrt{5}(4)+2\sqrt{5}(-3\sqrt{5})
\\\\=
12-9\sqrt{5}+8\sqrt{5}-6(5)
\\\\=
12-9\sqrt{5}+8\sqrt{5}-30
\\\\=
(12-30)+(-9\sqrt{5}+8\sqrt{5})
\\\\=
-18-\sqrt{5}
.\end{array}
