Answer
vertical asymptotes $x=-2$ and $x=7$,
horizontal asymptote $none$,
oblique asymptote $y=x+5$.
Work Step by Step
Step 1. Factor the function $R(x)=\frac{x^3+1}{x^2-5x-14}=\frac{x^3+1}{(x-7)(x+2)}$,
Step 2. Perform long division without factoring or synthetic divisions after factoring the denominator, $R(x)=\frac{x^3+1}{x^2-5x-14}=x+5+\frac{39x+71}{x^2-5x-14}$,
Step 3. we can find the vertical asymptotes $x=-2$ and $x=7$,
Step 4. we can find the horizontal asymptote $none$,
Step 5. we can find the oblique asymptote $y=x+5$.
