\[y=\frac{{x}^{2}+8}{-x-3}\]

Since the equation has $-x-3$ in the denominator, there is a vertical asymptote at $x=-3$.

Find the highest-degree terms in the numerator and the denominator.

Since the numerator has a higher degree than the denominator, there are no horizontal asymptotes.

From above, since the numerator is one degree higher than the denominator, there is a slant asymptote. Find it using polynomial long division.

Polynomial Division: Divide ${x}^{2}+8$ by $-x-3$.

Rewrite the expression using the above.

Take the quotient part only, which is $-x+3$. Therefore, the slant asymptote is: