$(2,x)$ is not a principal ideal of the ring $\mathbb{Z}[x]$

Question

Let $R=\mathbb{Z}[x]$ be a ring of integers with usual addition and multiplication. Consider $$I:=(2,x):=\Big\{2\alpha(x)+x\beta(x)\ :\ \alpha(x),\beta(x)\in \mathbb{Z}[x]\Big\}.$$ I have to prove that the above one is ideal but not a principal ideal.

I have proved that

1. $(I,+)$ is a group and
2. $\forall f(x)\in \mathbb{Z}[x],\ \forall\ (2\alpha(x)+x\beta(x))=g(x)\in I, f(x)g(x) \in I,g(x),f(x)\in I.$

But I am unable to prove that it is not a principal ideal.

Answer 1

Suppose that $(2,x)=(p)$, this implies that $2=pq$ this implies that $p$ and $q$ are constant and $p$ divides $2$. Since $(2,x)\neq Z[X]$, $p=2$. You also have $x=2q'$. Impossible since the coefficients of $2q$ are even.