Show that ⟨2,x⟩\langle 2,x \rangle is not a principal ideal in Z[x]\mathbb Z [x]

I think it’s relatively easy to see that $I=\langle 2,x \rangle = \{a_nx^n+\dots+a_1x+a_0; a_0\text{ is even}\}$.

Now, suppose that $I=\langle f(x) \rangle$ for some $f(x)\in I$.

If $f(x)$ is a constant polynomial, then $\langle f(x) \rangle$ contains only polynomials with even coefficients, and we do not get $x$.

If $f(x)$ is of degree at least $1$, then non-zero polynomials in $\langle f(x) \rangle$ have degree at least $1$, and we do not get $2$.

So $I$ is not of the form $\langle f(x) \rangle$.