Answer
Improper;
Proper Form: $3+\frac{1}{x^2-1}$
Work Step by Step
A rational expression is proper if and only if the degree of its numerator's polynomial is less than the degree of its denominator's polynomial.
Here, the degree of the numerator is $2$ while the degree of the denominator is also $2$
Hence, the rational expression is improper.
Make the rational expression proper by rewriting then simplify:
\begin{align*}
\frac{3x^2-2}{x^2-1}&=\frac{3x^2-3+1}{x^2-1}\\
&=\dfrac{3(x^2-1)+1}{x^2-1}\\
&=\frac{3(x^2-1)}{x^2-1}+\frac{1}{x^2-1}\\
&=3+\frac{1}{x^2-1}
\end{align*}
