So I'm trying to prove that if there exists a $5 \times 5$ matrix $Q$ such that
$$Q \succeq0,\,\, a_{l-1} = \sum\limits_{i+j=l} Q_{ij} , l=1,\ldots,5$$
then there exists a fourth degree polynomial $p(x)$ such that $$ p(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 \geq 0\,\, \forall x\in \mathbb{R} $$
I just managed to finish proving the converse and I have some intuition that the PSD property of Q somehow implies non-negativity of $p(x)$ but I'm not sure how to proceed from there.