Answer
$ \displaystyle \frac{3x^{2}-2}{x^{2}-1}$ is improper,
$ \displaystyle \frac{3x^{2}-2}{x^{2}-1}=3+\frac{1}{x^{2}-1}$
Work Step by Step
The rational expression $\displaystyle \frac{P}{Q}$ is called proper if
the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Otherwise, the rational expression is called improper.
---
The degree of the numerator, $P(x)=3x^{2}-2, \quad $is $2$, and
the degree of the denominator$ , Q(x)=x^{2}-1, \quad $is $2$.
The rational expression is improper.
To make it proper,
$ \displaystyle \frac{3x^{2}-2}{x^{2}-1} = \frac{3x^{2}-3+3-2}{x^{2}-1}=\frac{3(x^{2}-1)}{x^{2}-1}+\frac{3-2}{x^{2}-1}$
$ \displaystyle \frac{3x^{2}-2}{x^{2}-1} =3+\frac{1}{x^{2}-1}$
