Answer
$10-22\sqrt{3}$
Work Step by Step
Using $(a+b)(c+d)=ac+ad+bc+bd$, the given expression, $
\left( \sqrt{6}-4\sqrt{2} \right)\left( 3\sqrt{6}+\sqrt{2} \right)
,$ is equivalent to
\begin{array}{l}\require{cancel}
\sqrt{6}(3\sqrt{6})+\sqrt{6}(\sqrt{2})-4\sqrt{2}( 3\sqrt{6})-4\sqrt{2}(\sqrt{2})
.\end{array}
Using the properties of radicals, the expression above simplifies to
\begin{array}{l}\require{cancel}
3\sqrt{6(6)}+\sqrt{6(2)}-4(3)\sqrt{2(6)}-4\sqrt{2(2)}
\\\\=
3\sqrt{36}+\sqrt{12}-12\sqrt{12}-4\sqrt{4}
\\\\=
3\sqrt{36}+\sqrt{4\cdot3}-12\sqrt{4\cdot3}-4\sqrt{4}
\\\\=
3\sqrt{(6)^2}+\sqrt{(2)^2\cdot3}-12\sqrt{(2)^2\cdot3}-4\sqrt{(2)^2}
\\\\=
3(6)+2\sqrt{3}-12(2)\sqrt{3}-4(2)
\\\\=
18+2\sqrt{3}-24\sqrt{3}-8
\\\\=
(18-8)+(2\sqrt{3}-24\sqrt{3})
\\\\=
10-22\sqrt{3}
.\end{array}