Answer
$-\sqrt{2}+2\sqrt{3}$
Work Step by Step
Multiplying the numerator and the denominator by the conjugate of the denominator, the given expression is equivalent to:
\begin{align*}\require{cancel}
&
=\dfrac{4+\sqrt{6}}{\sqrt{2}+\sqrt{3}}\cdot\dfrac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}
\\\\&=
\dfrac{(4+\sqrt{6})(\sqrt{2}-\sqrt{3})}{(\sqrt{2})^2-(\sqrt{3})^2}
&\left( \text{use }(a+b)(a-b)=a^2-b^2 \right)
\\\\&=
\dfrac{(4+\sqrt{6})(\sqrt{2}-\sqrt{3})}{2-3}
\\\\&=
\dfrac{(4+\sqrt{6})(\sqrt{2}-\sqrt{3})}{-1}
\\\\&=
-(4+\sqrt{6})(\sqrt{2}-\sqrt{3})
\\&=
-[4(\sqrt{2})+4(-\sqrt{3}+\sqrt{6}(\sqrt{2})+\sqrt{6}(-\sqrt{3})]
&\left( \text{use FOIL} \right)
\\&=
-[4\sqrt{2}-4\sqrt{3}+\sqrt{12}-\sqrt{18}]
\\&=
-[4\sqrt{2}-4\sqrt{3}+\sqrt{4\cdot3}-\sqrt{9\cdot2}]
&\left( \text{extract perfect powers of index} \right)
\\&=-[4\sqrt{2}-4\sqrt{3}+2\sqrt{3}-3\sqrt{2}]
\\&=
-4\sqrt{2}+4\sqrt{3}-2\sqrt{3}+3\sqrt{2}
\\&=
(-4\sqrt{2}+3\sqrt{2})+(4\sqrt{3}-2\sqrt{3})
&\left( \text{combine like terms} \right)
\\&=
-\sqrt{2}+2\sqrt{3}
\end{align*}
Hence, the rationalized-denominator form of the given expression is $
-\sqrt{2}+2\sqrt{3}
$.
