Task B-2.5. Determine the values of the real parameter $a$ such that the inequality

$$
(a+1) x^{2}-3(a+1) x+a<0
$$

is true for every $x \in \mathbb{R}$.
Solution. It must hold that $a+1<0$ and $D=9(a+1)^{2}-4(a+1) a<0$, that is,

$$
(a+1)(5 a+9)<0
$$

From the first inequality, $a \in(-\infty,-1)$, and from the second, $a \in\left(-\frac{9}{5},-1\right)$.

Therefore, the given inequality is true for every $x \in \mathbb{R}$ if $a \in\left(-\frac{9}{5},-1\right)$.