8.44 Find all values of $a$ for which the expression

$$
\sqrt{(a+1) x^{2}-2(a-1) x+3 a-3}
$$

makes sense for any $x \in \boldsymbol{R}$.
8.44 The expression is defined under the condition $(a+1) x^{2}-2(a-1) x+3 a-3 \geq 0$.

1) Let $a+1=0$, i.e., $a=-1$. Then the inequality

will take the form $4 x-6 \geq 0$, from which $x \geq \frac{3}{2}$, i.e., it is not satisfied for all $x$.

2) Let $a+1 \neq 0$. Then the inequality (1) is satisfied for any $x$ if the discriminant of the quadratic trinomial is non-positive and $a+1>0$. Thus, we have the system of inequalities

$$
\left\{\begin{array}{l}
4(a-1)^{2}-4(a+1) \cdot 3(a-1) \leq 0, \\
a+1 > 0,
\end{array} \text { or } \left\{\begin{array}{l}
2(a-1)(a+2) \geq 0, \\
a+1>0
\end{array}\right.\right.
$$

from which we obtain that $1 \leq a<\infty$.

Answer: $1 \leq a<\infty$.