Answer
$ \displaystyle \frac{5x^{3}+2x-1}{x^{2}-4}$ is improper$, $
$\displaystyle \frac{5x^{3}+2x-1}{x^{2}-4}=5x+\frac{22x-1}{x^{2}-4}$
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.
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The degree of the numerator, $P(x)=5x^{3}+2x-1, \quad $is $3$, and
the degree of the denominator$ , Q(x)=x^{2}-4, \quad $is $2$.
The rational expression is improper.
To make it proper,
$\displaystyle \frac{5x^{3}+2x-1}{x^{2}-4} = \frac{5x(x^{2}-4)+20x+2x-1}{x^{2}-4}=\\\\\\ =\dfrac{5x(x^{2}-4)}{x^{2}-4}+\dfrac{22x-1}{x^{2}-4}$
$=5x+\displaystyle \frac{22x-1}{x^{2}-4}$
