Answer
V.A. $x=-2, x=7$, O.A. $y=x+5$.
Work Step by Step
Step 1. Given $y=\frac{x^3+1}{x^2-5x-14}=\frac{(x+1)(x^2-x+1)}{(x+2)(x-7)}$
Step 2. Use synthetic divisions (two steps) or long division (see figure), we have $y=x+5+\frac{39x+71}{x^2-5x-14}$
Step 3. We can find its vertical asymptote(s) V.A. $x=-2, x=7$, horizontal asymptote(s) H.A. $none$, oblique asymptote(s) O.A. $y=x+5$.
