Every positive polynomial is above a completely Q-factorized positive polynomial?

Let $P$ be a unitary polynomial with rational coefficients in one variable $x$, such that $P(x) \geq 0$ for all $x \in \mathbb R$. Then $P$ is of even degree, say $2d$. Is it true that there always exist $d$ rational numbers $q_1,q_2, \ldots ,q_d$ such that

$$P(x) \geq \bigg( \prod_{k=1}^{d} (x-q_k)^2\bigg),$$

for all $x\in \mathbb R$ ?

No. It's not even true for $d=1$: $(x-\sqrt{2})^2$ is a counterexample.