Answer
Vertical asymptotes: $x=1$ and $x=-1$
Horizontal asymptote: $y=0$
Work Step by Step
$r(x)=\dfrac{3x^{2}+5x}{x^{4}-1}$
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:
$x^{4}-1=0$
$x^{4}=1$
$\sqrt[4]{x^{4}}=\sqrt[4]{1}$
$x=\pm1$
Horizontal asymptote
Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote of this function is $y=0$