Answer
$0$
Work Step by Step
Recall that if $ f $ is a polynomial function, then we have $\lim_\limits{x\to a}f(x)=f(a)$.
In order to to find the limit, we will plug $ a $ into the function and then simplify.
$\lim_\limits{x\to -1} \dfrac{x^3+2x^2+x}{x^4+x^3+2x+2}=\dfrac{\lim_\limits{x\to -1}
x(x+1)^2 }{\lim_\limits{x\to -1} (x+1)(x^3+1)}=\dfrac{-1 \cdot 0}{(-1)^3 +2}=0$
