Show that the ideal $I = 3\mathbb Z[x] + (x^3-x^2+2x-1)\mathbb Z[x]$ is not a principal ideal of $\mathbb Z[x]$

Question

Show that the ideal $I = 3\mathbb Z[x] + (x^3-x^2+2x-1)\mathbb Z[x]$ is not a principal ideal of $\mathbb Z[x]$

Intuition: Assume that $$3\mathbb Z[x] + (x^3-x^2+2x-1)\mathbb Z[x]=<p(x)>$$ for $p(x) \in Z[x]$

So$$3\mathbb Z[x] + (x^3-x^2+2x-1)\mathbb Z[x]=p(x)Z[x]$$

Since $3\in p(x)Z[x]$ $$3=p(x)q(x) $$ where $q(x)\in Z[x]$

Which implies $p(x)=-3,-1,1$ or $3$. Since this is a contradiction $$3\mathbb Z[x] + (x^3-x^2+2x-1)\mathbb Z[x]\ne<p(x)>$$ for any $p(x)\in Z[x]$

Answer 1

Your intuition is pretty much a proof already; indeed if the ideal is principal then its generator must divide $3$, hence it must be one of $\{-3,-1,1,3\}$. Clearly $\pm3$ does not divide $x^3-x^2+2x-1$ and to show that it cannot be generated by $\pm1$ it suffices to show that the ideal is not the entire ring $\Bbb{Z}[x]$.