\(\frac{\sqrt{2+x}+\sqrt{2-x}}{\sqrt{2+x}-\sqrt{2-x}}=\sqrt{2}\Rightarrow\left(\frac{\sqrt{2+x}+\sqrt{2-x}}{\sqrt{2+x}-\sqrt{2-x}}\right)^2=2\)
\(\Rightarrow\frac{4+2\sqrt{4-x^2}}{4-2\sqrt{4-x^2}}=2\Rightarrow\frac{2+\sqrt{4-x^2}}{2-\sqrt{4-x^2}}=2\)
\(\Rightarrow2+\sqrt{4-x^2}=4-2\sqrt{4-x^2}\)
\(\Rightarrow3\sqrt{4-x^2}=2\Rightarrow\sqrt{4-x^2}=\frac{2}{3}\)
\(\Rightarrow4-x^2=\frac{4}{9}\Rightarrow x^2=\frac{32}{9}\Rightarrow x=\frac{4\sqrt{2}}{3}\)
\(\Rightarrow\frac{x+2}{x-2}=-17-12\sqrt{2}\)
