51. $$-\sqrt{20}(\frac{2-\sqrt{5}}{2+\sqrt{5}})^{2}+\frac{62}{\sqrt{5}-6}$$

51. Решим пример:

• $$-\sqrt{20}(\frac{2-\sqrt{5}}{2+\sqrt{5}})^{2}+\frac{62}{\sqrt{5}-6} = -2\sqrt{5} \cdot (\frac{2-\sqrt{5}}{2+\sqrt{5}})^{2}+\frac{62}{\sqrt{5}-6} = -2\sqrt{5} \cdot (\frac{(2-\sqrt{5})^{2}}{(2+\sqrt{5})^{2}})+\frac{62}{\sqrt{5}-6} = -2\sqrt{5} \cdot (\frac{4-4\sqrt{5}+5}{4+4\sqrt{5}+5})+\frac{62}{\sqrt{5}-6} = -2\sqrt{5} \cdot (\frac{9-4\sqrt{5}}{9+4\sqrt{5}})+\frac{62}{\sqrt{5}-6} = -2\sqrt{5} \cdot (\frac{(9-4\sqrt{5})(9-4\sqrt{5})}{(9+4\sqrt{5})(9-4\sqrt{5})})+\frac{62}{\sqrt{5}-6} = -2\sqrt{5} \cdot (\frac{(9-4\sqrt{5})^{2}}{81-16 \cdot 5})+\frac{62}{\sqrt{5}-6} = -2\sqrt{5} \cdot (\frac{(9-4\sqrt{5})^{2}}{81-80})+\frac{62}{\sqrt{5}-6} = -2\sqrt{5} \cdot (9-4\sqrt{5})^{2}+\frac{62}{\sqrt{5}-6} = -2\sqrt{5} \cdot (81-72\sqrt{5}+16 \cdot 5)+\frac{62}{\sqrt{5}-6} = -2\sqrt{5} \cdot (81-72\sqrt{5}+80)+\frac{62}{\sqrt{5}-6} = -2\sqrt{5} \cdot (161-72\sqrt{5})+\frac{62}{\sqrt{5}-6} = -322\sqrt{5}+144 \cdot 5 + \frac{62}{\sqrt{5}-6} = -322\sqrt{5}+720 + \frac{62}{\sqrt{5}-6} = -322\sqrt{5}+720 + \frac{62(\sqrt{5}+6)}{(\sqrt{5}-6)(\sqrt{5}+6)} = -322\sqrt{5}+720 + \frac{62(\sqrt{5}+6)}{5-36} = -322\sqrt{5}+720 + \frac{62(\sqrt{5}+6)}{-31} = -322\sqrt{5}+720 - 2(\sqrt{5}+6) = -322\sqrt{5}+720 - 2\sqrt{5}-12 = -324\sqrt{5} + 708$$

Ответ:$$-324\sqrt{5} + 708$$