Contain does not imply divide

Question

Let $\mathbb{Z}[X]$ be the ring of polynomials over a single variable $X$. Let $(2)\subseteq(2,X)$ be ideals of $\mathbb{Z}[X]$. I want to prove that it is impossible to write the ideal $(2)$ as a product $(2,X)I=(2)$, where $I\subseteq \mathbb{Z}[X]$ is an ideal. I tried so far to write $2=\sum_{i=1}^n(2\lambda_i+X\mu_i)b_i$, where $\lambda_i,\mu_i\in \mathbb{Z}[X]$ and $b_i\in I$ so that I could find a contradiction. Nonetheless, I don't know how to continue. Any clues?