Answer
$\sqrt{6}-\sqrt{2}+\sqrt{3}-1$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
(\sqrt{2}+1)(\sqrt{3}-1)
,$ use FOIL and the properties 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}
\sqrt{2}(\sqrt{3})+\sqrt{2}(-1)+1(\sqrt{3})+1(-1)
\\\\=
\sqrt{2}(\sqrt{3})-\sqrt{2}+\sqrt{3}-1
.\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}
\sqrt{2(3)}-\sqrt{2}+\sqrt{3}-1
\\\\=
\sqrt{6}-\sqrt{2}+\sqrt{3}-1
.\end{array}
