Answer
The rational expression is improper. To make it proper,
$ \displaystyle \frac{3x^{4}+x^{2}-2}{x^{3}+8}=3x+\frac{x^{2}-24x-2}{x^{3}+8}$
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)=3x^{4}+x^{2}-21, \quad $is $4$, and
the degree of the denominator$ , Q(x)=x^{3}+8, \quad $is $3$.
The rational expression is improper.
To make it proper,
$ \displaystyle \frac{3x^{4}+x^{2}-2}{x^{3}+8} = \frac{3x(x^{3}+8)-24x+x^{2}-2}{x^{3}+8}$
$=\displaystyle \frac{3x(x^{3}+8)}{x^{3}+8}+\frac{x^{2}-24x-2}{x^{3}+8}$
$=3x+\displaystyle \frac{x^{2}-24x-2}{x^{3}+8}$
