Answer
$-24-7\sqrt{6}$
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, $
(2\sqrt{3}+3\sqrt{2})(\sqrt{3}-5\sqrt{2})
,$ simplifies to
\begin{array}{l}\require{cancel}
2\sqrt{3}(\sqrt{3})+2\sqrt{3}(-5\sqrt{2})+3\sqrt{2}(\sqrt{3})+3\sqrt{2}(-5\sqrt{2})
\\\\=
2(3)-10\sqrt{6}+3\sqrt{6}-15(2)
\\\\=
6-10\sqrt{6}+3\sqrt{6}-30
\\\\=
(6-30)+(-10\sqrt{6}+3\sqrt{6})
\\\\=
-24-7\sqrt{6}
.\end{array}
