\(\dfrac{\left(\sqrt{5}+2\right)^2-8\sqrt{5}}{2\sqrt{5}-4}\)
\(=\dfrac{5+4\sqrt{5}+4-8\sqrt{5}}{2\sqrt{5}-4}\)
\(=\dfrac{9-4\sqrt{5}}{2\sqrt{5}-4}\)
\(=\dfrac{\left(9-4\sqrt{5}\right)\left(2\sqrt{5}+4\right)}{4}\)
\(=\dfrac{\left(9-4\sqrt{5}\right)\cdot2\left(\sqrt{5}+2\right)}{4}\)
\(=\dfrac{\left(9-4\sqrt{5}\right)\left(\sqrt{5}+2\right)}{2}\)
\(=\dfrac{9\sqrt{5}+18-20-8\sqrt{5}}{2}\)
\(=\dfrac{\sqrt{5}-2}{2}\)
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