\[\left\{ \begin{matrix} x^{2} + 2y^{2} + \frac{5}{x^{2} + 2y^{2} + 1} = 3 \\ x^{2} - y^{2} - 3x + 5y = 10\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[x^{2} + 2y^{2} + 1 + \frac{5}{x^{2} + 2y^{2} + 1} = 4\]
\[t = x^{2} + 2y^{2} + 1\]
\[t + \frac{5}{t} = 4\ \ \ \ \ \ | \cdot t\]
\[t^{2} + 5 = 4t\]
\[t^{2} - 4t + 5 = 0\]
\[D = ( - 4)^{2} - 4 \cdot 1 \cdot 5 =\]
\[= 16 - 20 = - 4 < 0 \Longrightarrow\]
\[\Longrightarrow нет\ решений.\]
\[\Longrightarrow ч.т.д.\]