#### Answer

$2+\sqrt{6}-2\sqrt{3}-3\sqrt{2}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
(1-\sqrt{3})(2+\sqrt{6})
,$ use the FOIL Method and the laws of radicals.
$\bf{\text{Solution Details:}}$
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}
1(2)+1(\sqrt{6})-\sqrt{3}(2)-\sqrt{3}(\sqrt{6})
\\\\=
2+\sqrt{6}-2\sqrt{3}-\sqrt{3}(\sqrt{6})
.\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}
2+\sqrt{6}-2\sqrt{3}-\sqrt{3(6)}
\\\\=
2+\sqrt{6}-2\sqrt{3}-\sqrt{18}
\\\\=
2+\sqrt{6}-2\sqrt{3}-\sqrt{9\cdot2}
\\\\=
2+\sqrt{6}-2\sqrt{3}-\sqrt{(3)^2\cdot2}
\\\\=
2+\sqrt{6}-2\sqrt{3}-3\sqrt{2}
.\end{array}