If $x=\frac{\sqrt{5}-2}{\sqrt{5}+2}$, then the value of $x^2+x^{-2}$ is:

322

If $x=\frac{\sqrt{5}-2}{\sqrt{5}+2}$

By rationalization 

$x=\frac{\sqrt{5}-2}{\sqrt{5}+2}$ × $\frac{\sqrt{5}-2}{\sqrt{5}-2}$ = 9 + 4\(\sqrt {5}\)

x² = (9 + 4\(\sqrt {5}\))² = 161 + 72\(\sqrt {5}\)

x^-2 = 161 - 72\(\sqrt {5}\)

$x^2+x^{-2}$ =  161 + 72\(\sqrt {5}\) + 161 - 72\(\sqrt {5}\) = 322