Ideal in $\mathbb Z[x]$ which is not two-generated
Solution 1:
Note that $(4, 2x, x^2)=(2,x)^2$. Set $J=(2,x)$. We have $\mathbb Z[x]/J\simeq\mathbb F_2$ (the field with two elements). Show that $4, 2x, x^2$ are linearly independent in $J^2/J^3$ over $\mathbb F_2$, so $\dim_{\mathbb F_2}J^2/J^3\ge3$. If $J^2$ is two-generated, then the quotient $J^2/J^3$ is also two-generated, a contradiction.