Pinter, in his book "A book of Abstract Algebra" asks the reader to divide $x^3+2$ by $2x^2+3x+4$ in $\mathbb{Z}[x]$. But I do not see how it can be done. Suppose that it could be done. Then there would exist polynomials $px+q$ and $p'x+q'$ in $\mathbb{Z}[x]$ such that $$ x^3+2=(px+q)(2x^2+3x+4)+(p'x+q')$$
This would imply that $2p=1$ but there is no $p \in \mathbb{Z}$ such that $2p=1$. Thus, it is not possible to do the usual euclidean division. Is this question incorrect or am I going wrong somewhere?
The question is correct, and your answer is correct as well. It is simply not possible to do that division in $\mathbb{Z}[x]$.