Answer
$0$.
Work Step by Step
As we can see from the attached graph, as $x$ gets closer and closer to $-1$ from the left and the right, the value of the function (the $v$y-value) gets closer and closer to $0$. Thus the limit is
$\displaystyle \lim_{x\to-1}\frac{x^3+2x^2+x}{x^4+x^3+2x+2}=0$
