such that $\displaystyle -2 < x^3 - 2x^2 + x - 2 < 0 $.
Concentrate on one side of the inequality at a time.
$\displaystyle -2 < x^3-2x^2+x-2$
$\displaystyle \implies\ 0<x^3-2x^2+x=x(x^2-2x+1)=x(x-1)^2$
Solive this, then go on to the next inequality.
$\displaystyle x^3-2x^2+x-2<0$
Solve this (hint: $\displaystyle x-2$ is a factor of the LHS). Then combine the two solutions.
$\displaystyle -2$ divided by $\displaystyle x^2+1$ is not $\displaystyle -2.$