#### Answer

$-19+\sqrt{77}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
(\sqrt{7}-\sqrt{11})(2\sqrt{7}+3\sqrt{11})
,$ use FOIL and the properties of radicals. Then combine like terms.
$\bf{\text{Solution Details:}}$
Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\sqrt{7}(2\sqrt{7})+\sqrt{7}(3\sqrt{11})-\sqrt{11}(2\sqrt{7})-\sqrt{11}(3\sqrt{11})
\\\\=
1(2)(\sqrt{7})^2+1(3)\sqrt{7}(\sqrt{11})-1(2)\sqrt{11}(\sqrt{7})-1(3)(\sqrt{11})^2
\\\\=
2(7)+3\sqrt{7}(\sqrt{11})-2\sqrt{11}(\sqrt{7})-3(11)
\\\\=
14+3\sqrt{7}(\sqrt{11})-2\sqrt{11}(\sqrt{7})-33
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
14+3\sqrt{7(11)}-2\sqrt{11(7)}-33
\\\\=
14+3\sqrt{77}-2\sqrt{77}-33
.\end{array}
By combining like terms, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(14-33)+(3\sqrt{77}-2\sqrt{77})
\\\\=
-19+\sqrt{77}
.\end{array}