\[x^{2} + 10x + 4 = 0\]
\[x_{1} + x_{2} = - 10;\]
\[x_{1} \cdot x_{2} = 4;\]
\[\left( x_{1} + x_{2} \right)^{2} = 100;\]
\[x_{1}^{2} + x_{2}^{2} = \left( x_{1} + x_{2} \right)^{2} - 2x_{1}x_{2} =\]
\[= 100 - 2 \cdot 4 = 100 - 8 = 92.\]
\[Ответ:92.\]