Answer
$\dfrac{5+3\sqrt{2}}{7}$
Work Step by Step
Rationalizing the denominator of $
\dfrac{2\sqrt{3}+\sqrt{6}}{4\sqrt{3}-\sqrt{6}}
$ results to
\begin{array}{l}
\dfrac{2\sqrt{3}+\sqrt{6}}{4\sqrt{3}-\sqrt{6}}
\cdot
\dfrac{4\sqrt{3}+\sqrt{6}}{4\sqrt{3}+\sqrt{6}}
\\\\=
\dfrac{(2\sqrt{3})(4\sqrt{3})+(2\sqrt{3})(\sqrt{6})+(\sqrt{6})(4\sqrt{3})+(\sqrt{6})(\sqrt{6})}{(4\sqrt{3})^2-(\sqrt{6})^2}
\\\\=
\dfrac{8\cdot3+2\sqrt{18}+4\sqrt{18}+6}{16\cdot3-6}
\\\\=
\dfrac{24+6\sqrt{18}+6}{42}
\\\\=
\dfrac{30+6\sqrt{9\cdot2}}{42}
\\\\=
\dfrac{30+18\sqrt{2}}{42}
\\\\=
\dfrac{5+3\sqrt{2}}{7}
.\end{array}
