Answer
$\sqrt{6}+\sqrt{2}-4\sqrt{3}-4$
Work Step by Step
Using the properties of radicals, the given expression, $
(\sqrt{2}-4)(\sqrt{3}+1)
,$ simplifies to
\begin{array}{l}\require{cancel}
(\sqrt{2}-4)(\sqrt{3}+1)
\\\\=
\sqrt{2}(\sqrt{3})+\sqrt{2}(1)-4(\sqrt{3})-4(1)
\\\\=
\sqrt{2(3)}+\sqrt{2}-4\sqrt{3}-4
\\\\=
\sqrt{6}+\sqrt{2}-4\sqrt{3}-4
.\end{array}
