9.115. Find the values of $m$ for which the inequality

$$
\frac{x^{2}-8 x+20}{m x^{2}+2(m+1) x+9 m+4}<0
$$

is satisfied for all real values of $x$.
## Solution.

Since $x^{2}-8 x+20>0$ for $x \in R$, it is necessary that

$$
m x^{2}+2(m+1) x+9 m+40, \\
m<0
\end{array} \Leftrightarrow m<-\frac{1}{2}\right.\right.
$$

Answer: $m \in\left(-\infty ;-\frac{1}{2}\right)$.