#### Answer

$1-5\sqrt[]{7}$

#### Work Step by Step

Using the FOIL method and the properties of radicals, the given expression, $
(\sqrt[]{7}-6)(\sqrt[]{7}+1)
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt[]{7}(\sqrt[]{7})+\sqrt[]{7}(1)-6(\sqrt[]{7})-6(1)
\\\\=
7+\sqrt[]{7}-6\sqrt[]{7}-6
\\\\=
(7-6)+(\sqrt[]{7}-6\sqrt[]{7})
\\\\=
1-5\sqrt[]{7}
.\end{array}