Let's consider the polynomial $ f\left( x \right) = \left( {x^2 + 2} \right)\prod\limits_{i = - k}^k {\left( {x - 2i} \right) + 2 \in {\Bbb Q}\left[ x \right]} $ . Let's suppose that $ p = 2k + 3 \geqslant 5 $ is prime.
Prove the following:
$i)$ Prove that $f$ is irreducible and of degree $p$
$ii)$ Prove that p has exactly $p-2$ real zeros. I have no idea how to prove this, maybe with einsenstein and considering the derivate, but how? :/