Find an Ideal of $\mathbb{Z}+x \mathbb{Q}[ x ]$ that is NOT principal 
The ring $\mathbb{Z}+x \mathbb{Q}[ x ]$ cannot be a principal ideal domain since it is not a unique factorization domain. Find an ideal of $\mathbb{Z}+x \mathbb{Q}[ x ]$ that is not principal. 

My book gives no examples of how to show an ideal is not principal. I'm pretty sure if I let $I=(2,1/2 x)$ then I can show it's not principal. 
I'm pretty confident that the best route to go is to do a proof by contradiction. But how do I start? What is my initial assumption? 
Thanks!

Another ideal which is not principal is $X\mathbb Q[X]$. (In fact, it is not even finitely generated.) Suppose the contrary, and let $f\in X\mathbb Q[X]$, say $f=Xg$ with $g\in\mathbb Q[X]$, such that $X\mathbb Q[X]=(f)$. Then since $X(\frac12g)\in X\mathbb Q[X]$ there is $h\in R$ such that $X(\frac12g)=fh$, that is, $X(\frac12g)=Xgh$. It follows $h=\frac12\notin\mathbb Z$, a contradiction. 
