Answer
$-11+\sqrt{21}$
Work Step by Step
Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the given expression, $
(\sqrt{3}-\sqrt{7})(\sqrt{3}+2\sqrt{7})
,$ is equivalent to
\begin{align*}
&
\sqrt{3}(\sqrt{3})+\sqrt{3}(2\sqrt{7})-\sqrt{7}(\sqrt{3})-\sqrt{7}(2\sqrt{7})
\\&=
(\sqrt{3})^2+2\sqrt{3(7)}-\sqrt{7(3)}-2(\sqrt{7})^2
\\&=
3+2\sqrt{21}-\sqrt{21}-2(7)
\\&=
3+2\sqrt{21}-\sqrt{21}-14
\\&=
(3-14)+(2\sqrt{21}-\sqrt{21})
\\&=
-11+\sqrt{21}
.\end{align*}
Hence, the simplified form of the given expression is $
-11+\sqrt{21}
$.
