Answer
Vertical asymptotes: $x=2$ and $x=-2$
Horizontal asymptote: None
Work Step by Step
$r(x)=\dfrac{x^{3}+3x^{2}}{x^{2}-4}$
Vertical asymptotes
A rational function has vertical asymptotes where the function is undefined, that is, where the denominator is zero.
Set the denominator equal to 0 and solve for x to find the vertical asymptotes of this function:
$x^{2}-4=0$
$x^{2}=4$
$\sqrt{x^{2}}=\sqrt{4}$
$x=\pm2$
Horizontal asymptote
The degree of the numerator is greater than the degree of the denominator. This function has no horizontal asymptote.