Algebra

$Compute\ \frac{\sqrt{3} - 4 \sqrt{5}}{(\sqrt{3})^2 + (\sqrt{2})^2}\\ \frac{\sqrt{3} - 4 \sqrt{5}}{(\sqrt{3})^2 + (\sqrt{2})^2}\\ =\frac{\sqrt{3} - 4 \sqrt{5}}{(\sqrt{3})^2 +\ (\sqrt{2})^2}\cdot \frac{(\sqrt{3})^2 -\ (\sqrt{2})^2}{(\sqrt{3})^2 -\ (\sqrt{2})^2}\\ =\frac{3\sqrt{3}-2\sqrt{3}-12\sqrt{5}+8\sqrt{5}}{9-4}\\ \color{blue}=\frac{\sqrt{3}-4\sqrt{5}}{5}$

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