1. What relation that does not depend on $m$ exists between the solutions of the equation

$$
\left(x^{2}-6 x+5\right)+m\left(x^{2}-5 x+6\right)=0 ?
$$
Solution. If $m \neq -1$, then the given equation is equivalent to the equation

$$
x^{2}-\frac{5 m+6}{m+1} x+\frac{6 m+5}{m+1}=0
$$

and has solutions $x_{1}$ and $x_{2}$ for which

$$
\begin{aligned}
& x_{1}+x_{2}=\frac{5 m+6}{m+1}=5+\frac{1}{m+1} \\
& x_{1} x_{2}=\frac{6 m+5}{m+1}=6-\frac{1}{m+1}
\end{aligned}
$$

If we add the last two equations, we get

$$
x_{1}+x_{2}+x_{1} x_{2}=11
$$