Answer
$$-\frac{2}{3}$$
Work Step by Step
Since we have
$$
\lim _{x \rightarrow-1} \frac{x^{4}+2 x+1}{x^{5}-2 x-1}=\frac{1-2 +1}{-1+2 -1}=\frac{0}{0}
$$
then we can apply L’Hôpital’s Rule as follows
$$
\lim _{x \rightarrow-1} \frac{x^{4}+2 x+1}{x^{5}-2 x-1}=\lim _{x \rightarrow-1} \frac{4x^{3}+2 }{5x^{4}-2 }=\frac{-4 +2 }{5 -2 }=-\frac{2}{3}.
$$
