Find the horizontal and vertical asymptote(s) of $LaTeX: \displaystyle f(x) = \frac{x^{2} + 15 x + 56}{5 x^{2} + 20 x + 15}$ , if they exist.

The degree of the numerator is the same as the degree of the denominator, so the horizontal asymptote is the ratio of the leading coefficients $LaTeX: \displaystyle y=\frac{1}{5}$ . The vertical asymptotes are the zeros of the denominator. $LaTeX: \displaystyle 5 x^{2} + 20 x + 15=0 \iff 5 \left(x + 1\right) \left(x + 3\right)=0$ . The vertical asymptotes are $LaTeX: \displaystyle x=-3$ and $LaTeX: \displaystyle x=-1$ .