Inequality but with Polynomials

Consider all polynomials $P(x)$ with real coefficients satisfying $P(1) = \sqrt{2}$, $P(-1) = -\sqrt{2}$ and $$\big(P(x)\big)^2 \le x^6 + 1 \quad \text{for all } x \in \mathbb{R}.$$ How many such polynomials $P(x)$ are there?