Prove convergence of improper integral of hard to factor denominator

Question

I am trying to determine whether $$ \int_{-3}^{2} \frac{xdx}{x^3+2x+2}\,.$$ converges or diverges

Usually, for improper integrals where there is a vertical asymptote, I try to split the integral up around the asymptote, but I can't really do that here cause I cannot factor it. Any recommendations?

Thanks

Answer 1

Hint: The denominator is positive at $0$ and negative at $-1,$ hence must be $0$ somewhere in between.