HMMT 2011 U27

Find the number of polynomials $p(x)$ with integer coefficients satisfying $p(x) \geq \min \left\{2 x^{4}-6 x^{2}+1,4-5 x^{2}\right\}$ and $p(x) \leq \max \left\{2 x^{4}-6 x^{2}+1,4-5 x^{2}\right\}$ for all $x \in \mathbb{R}$.