Answer
$4+5\sqrt{42}$
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, $
(\sqrt{6}+2\sqrt{7})(3\sqrt{6}-\sqrt{7})
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt{6}(3\sqrt{6})+\sqrt{6}(-\sqrt{7})+2\sqrt{7}(3\sqrt{6})+2\sqrt{7}(-\sqrt{7})
\\\\=
3(6)-\sqrt{42}+6\sqrt{42}-2(7)
\\\\=
18-\sqrt{42}+6\sqrt{42}-14
\\\\=
(18-14)+(-\sqrt{42}+6\sqrt{42})
\\\\=
4+5\sqrt{42}
.\end{array}
