A polynomial reducible in $\mathbb{Z}$ and $\mathbb{R}$ but irreducible in $\mathbb{Q}$

Question

I'm trying to construct a polynomial $f(x)$ such that $f(x)$ is reducible over $\mathbb{Z}$ and $\mathbb{R}$ but irreducible in $\mathbb{Q}$.

So far I've constructed a polynomial $f(x) = x^2 + 2x + 2$, which I just want to know how to check if it is reducible or irreducible over $\mathbb{R}$ and $\mathbb{Z}$. Also I would like some advice on how to construct a polynomial that would satisfy the conditions given if such a polynomial exists.

Answer 1

I wonder if there is anything more basic than $2 x^2 - 4$.

Answer 2

I think I found a solution,$f(x) = 4x^3 + 12x^2 + 12x + 12$. Could someone verify if this polynomial satisfies the given conditions? I know $f(x)$ is cubic so is reducible over $\mathbb{R}$, and irreducible in $\mathbb{Q}$ by Eisenstein's Criteria.