Найдите значение выражения \(\displaystyle \frac{1}{2 + \sqrt5} - \frac{1}{\sqrt7 + 3} + \frac{3}{1 - \sqrt7} - \frac{10}{\sqrt5} + \sqrt5\).
\[\begin{gathered} \frac{1}{2 + \sqrt5} - \frac{1}{\sqrt7 + 3} + \frac{3}{1 - \sqrt7} - \frac{10}{\sqrt5} + \sqrt5 =\\= \frac{2 - \sqrt5}{(2 + \sqrt5)(2 - \sqrt5)} - \frac{\sqrt7 - 3}{(\sqrt7 + 3)(\sqrt7 - 3)} + \frac{3(1 + \sqrt7)}{(1 - \sqrt7)(1 + \sqrt7)} - \frac{10\sqrt5}{(\sqrt5)^2} + \sqrt5 =\\= \frac{2 - \sqrt5}{2^2 - (\sqrt5)^2} - \frac{\sqrt7 - 3}{(\sqrt7)^2 - 3^2} + \frac{3(1 + \sqrt7)}{1^2 - (\sqrt7)^2} - \frac{10\sqrt5}{5} + \sqrt5 =\\= \frac{2 - \sqrt5}{4 - 5} - \frac{\sqrt7 - 3}{7 - 9} + \frac{3(1 + \sqrt7)}{1 - 7} - 2\sqrt5 + \sqrt5 = \frac{2 - \sqrt5}{-1} - \frac{\sqrt7 - 3}{-2} + \frac{3(1 + \sqrt7)}{-6} - \sqrt5 =\\= -2 + \sqrt5 + \frac{\sqrt7}{2} - \frac{3}{2} - \frac{3}{6} - \frac{3\sqrt7}{6} - \sqrt5 = -2 + \frac{\sqrt7}{2} - \frac{3}{2} - \frac{1}{2} - \frac{\sqrt7}{2} = -4\end{gathered}\]
Ответ: -4