Answer
$\displaystyle \frac{41+5\sqrt{30}}{49}$
Work Step by Step
We lose the square roots in the denominator by applying the difference of squares formula:
$(a\sqrt{x}+b\sqrt{y})(a\sqrt{x}-b\sqrt{y})=(a\sqrt{x})^{2}-(b\sqrt{y})^{2}=a^{2}x-b^{2}y$
$\displaystyle \frac{2\sqrt{6}+\sqrt{5}}{3\sqrt{6}-\sqrt{5}}\color{red}{ \cdot\frac{3\sqrt{6}+\sqrt{5}}{3\sqrt{6}+\sqrt{5}} }\qquad$ (rationalize)
$=\displaystyle \frac{(2\sqrt{6}+\sqrt{5})(3\sqrt{6}+\sqrt{5})}{(3\sqrt{6})^{2}-(\sqrt{5})^{2}}$
... use FOIL for the numerator
$=\displaystyle \frac{6\cdot 6+2\sqrt{30}+3\sqrt{30}+5}{3^{2}\cdot 6-5}$
$=\displaystyle \frac{36+5\sqrt{30}+5}{54-5}$
$=\displaystyle \frac{41+5\sqrt{30}}{49}$
