Answer
$\text{improper}$;
$5x+\dfrac{22x-1}{x^{2}-4}$
Work Step by Step
A rational expression $\dfrac{A(x)}{B(x)}$ is said to be proper if the degree of polynomial in numerator is less than the degree of its denominator.If it does not happen , then it is said to be improper rational polynomial.
Here, the degree of the numerator $A(x)= 5x^3+2x+1$ is $3$ and the degree of the denominator ; $B(x)=x^2-4$ is $2$.
We see that the given rational expression is improper.
To make the rational expression proper, we will solve as:
$\dfrac{5x^{3}+2x-1}{x^{2}-4}=\dfrac{5x^{3}-20x+22x-1}{x^{2}-4} \\
\\=\dfrac{(5x^{3}-20x)+(22x-1)}{x^{2}-4} \\
\\=\dfrac{5x(x^{2}-4)}{x^{2}-4}+\dfrac{22x-1}{x^{2}-4}
\\=5x+\dfrac{22x-1}{x^{2}-4}$
