Answer
$0$
Work Step by Step
Factor each polynomial completely:
$\lim _{x\rightarrow -1}\dfrac {x^{3}+2x^{2}+x}{x^{4}+x^{3}+2x+2}
\\=\lim _{x\rightarrow -1}\dfrac {x\left( x^{2}+2x+1\right) }{x^{3}\left( x+1\right) +2\left( x+1\right) }
\\=\lim _{x\rightarrow -1}\dfrac {x\left( x+1\right) ^{2}}{\left( x+1\right) \left( x^{3}+2\right) }$
Cancel the common factors:
$\require{cancel}
\\=\lim _{x\rightarrow -1}\dfrac {x\left( x+1\right) ^\cancel{{2}}}{\cancel{\left( x+1\right)} \left( x^{3}+2\right)}
\\=\lim _{x\rightarrow -1}\dfrac {\left( x+1\right) \left( x\right) }{x^{3}+2}
\\=\dfrac {\left( -1+1\right) \times \left( -1\right) }{\left( -1\right) ^{3}+2}
\\=0$
