Irreducible polynomial over $\Bbb Q$

Question

I am trying to solve the following problem from few days, but yet not get any solution. $$\text{Let $f(x)\in \Bbb Q[x]$ be irreducible.}$$$$\text{ Then there is no $\alpha\in \Bbb C$ with $f(\alpha)=f(\alpha+1)=0$. }$$

I have tried using the fact that every polynomial over $\Bbb R$ can be written as product of quadratic and linear polynomial. But I don't know what I have to after that. Any help will be appreciated.

Answer 1

Hint: The polynomial $g(x) = f(x+1) - f(x)$ has $\alpha$ as a root.