MathNet 2011 Question 72

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