Ideals Containing Ideals

Question

I have the following problem that I am stuck on:

Consider the ideal $I=\left\langle X(X+1)^{2}\right\rangle$ of $\mathbf{Q}[X]$. How many ideals $J$ of $\mathbf{Q}[X]$ are there such that $I\subseteq J$?

I am completely lost on this problem. I feel like it should have a straightforward solution, but I am not seeing it. Any help is appreciated!

Answer 1

Hint: $\mathbb Q[X]$ is a principal ideal domain, i.e. every ideal in $\mathbb Q[X]$ is generated by a single element. If the ideal contains $I$, what could that element be?

Answer 2

Hint:

Use the Chinese remainder theorem: $$ \mathbf Q[X]/\bigl(X(X^2+1)^2\bigr)\simeq\mathbf Q[X]/(X)\times\mathbf Q[X]/\bigl((X^2+1)^2\bigl)\simeq\mathbf Q\times\mathbf Q[X]/\bigl((X^2+1)^2\bigl) $$