Let the polynomials be $p_1$, $p_2$, $p_3$, $p_4$. Consider the below array. \begin{align*} p_1+p_2&& p_1+p_2 && p_1+p_3&& p_2+p_3\\ p_1+p_3 && p_1+p_4 && p_1+p_4 && p_2+p_4\\ p_2+p_3&& p_2+p_4 && p_3+p_4 && p_3+p_4\end{align*} If we add each column, we get twice the sum of any three polynomials. So each entry of the array should be strictly >0 or <0, but the sum cannot. Thus the three entries in each column cannot all have the same sign. Consider the first column. WLOG set $p_1+p_2>0$, $p_1+p_3>0$, $p_2+p_3<0$. \begin{align*} p_1+p_2>0 \ (1) && p_1+p_2>0 && p_1+p_3>0 && p_2+p_3<0 \\ p_1+p_3>0 \ (2) && p_1+p_4 && p_1+p_4 && p_2+p_4\\ p_2+p_3<0 \ (3) && p_2+p_4 && p_3+p_4 && p_3+p_4\end{align*} Let $a$ be the sign of $p_1+p_4$, $b$ be the sign of $p_2+p_4$, $c$ be the sign of $p_3+p_4$. $p_1+p_2$, $p_1+p_4$, $p_2+p_4$ cannot all be the same sign, so at least one of $a$ and $b$ is (-). Similarly, at least one of $a$ and $c$ is (-), and at least one of $b$ and $c$ is (+). We see that $a$ cannot be (+), so $a$ is (-). Now we have $-p_1 < p_3 < -p_2$ from (2) and (3), so $p_1>p_2$. But $p_1+p_4<0$, so $p_2+p_4<0$. Similarly, $p_3+p_4<0$. But then notice that $b$ and $c$ are both (-), so we have a contradiction in the fourth column. Thus there do not exist four polynomials with the property.