If x = $\dfrac{4\sqrt{6}}{\sqrt{2} + \sqrt{3}}$, find the value of

$\dfrac{x + 2\sqrt{2}}{x - 2\sqrt{2}} + \dfrac{x + 2\sqrt{3}}{x - 2\sqrt{3}}$

Given,

$x = \dfrac{4\sqrt{6}}{\sqrt{2} + \sqrt{3}} \\[1em] \dfrac{x}{2\sqrt{2}} = \dfrac{\dfrac{4\sqrt{6}}{\sqrt{2} + \sqrt{3}}}{2\sqrt{2}} \\[1em] \therefore \dfrac{x}{2\sqrt{2}} = \dfrac{2\sqrt{3}}{\sqrt{2} + \sqrt{3}}$

By componendo and dividendo,

$\Rightarrow \dfrac{x + 2\sqrt{2}}{x - 2\sqrt{2}} = \dfrac{2\sqrt{3} + \sqrt{2} + \sqrt{3}}{2\sqrt{3} - \sqrt{2} - \sqrt{3}} \\[1em] \Rightarrow \dfrac{x + 2\sqrt{2}}{x - 2\sqrt{2}} = \dfrac{3\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \qquad \text{[….Eq 1]} \\[1.5em] \dfrac{x}{2\sqrt{3}} = \dfrac{\dfrac{4\sqrt{6}}{\sqrt{2} + \sqrt{3}}}{2\sqrt{3}} \\[1em] \therefore \dfrac{x}{2\sqrt{3}} = \dfrac{2\sqrt{2}}{\sqrt{2} + \sqrt{3}}$

By componendo and dividendo,

$\Rightarrow \dfrac{x + 2\sqrt{3}}{x - 2\sqrt{3}} = \dfrac{2\sqrt{2} + \sqrt{2} + \sqrt{3}}{2\sqrt{2} - \sqrt{2} - \sqrt{3}} \\[1em] \Rightarrow \dfrac{x + 2\sqrt{3}}{x - 2\sqrt{3}} = \dfrac{3\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}} \qquad \text{[….Eq 2]}$

Adding Eq 1 and 2,

$\Rightarrow \dfrac{x + 2\sqrt{2}}{x - 2\sqrt{2}} + \dfrac{x + 2\sqrt{3}}{x - 2\sqrt{3}} = \dfrac{3\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} + \dfrac{3\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}} \\[1em] \Rightarrow \dfrac{x + 2\sqrt{2}}{x - 2\sqrt{2}} + \dfrac{x + 2\sqrt{3}}{x - 2\sqrt{3}} = \dfrac{3\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} - \dfrac{3\sqrt{2} + \sqrt{3}}{\sqrt{3} - \sqrt{2}} \\[1em] \Rightarrow \dfrac{x + 2\sqrt{2}}{x - 2\sqrt{2}} + \dfrac{x + 2\sqrt{3}}{x - 2\sqrt{3}} = \dfrac{3\sqrt{3} + \sqrt{2} - 3\sqrt{2} - \sqrt{3}}{\sqrt{3} - \sqrt{2}} \\[1em] \Rightarrow \dfrac{x + 2\sqrt{2}}{x - 2\sqrt{2}} + \dfrac{x + 2\sqrt{3}}{x - 2\sqrt{3}} = \dfrac{2\sqrt{3} - 2\sqrt{2}}{\sqrt{3} - \sqrt{2}} \\[1em] \Rightarrow \dfrac{x + 2\sqrt{2}}{x - 2\sqrt{2}} + \dfrac{x + 2\sqrt{3}}{x - 2\sqrt{3}} = \dfrac{2(\sqrt{3} - \sqrt{2})}{\sqrt{3} - \sqrt{2}} = 2. \\[1em]$

Hence, the required value is 2.