Actually, it has nothing to do with geometry .. Let $r_i(t),\ i\in \overline{1,4}$ be the four real roots (arranged in increasing order from $i=1$ to $4$) of the polynomial $P_t(x)=P(x)-t,\ t\in \mathbb R$. We have to show that for all $t$ for which $P_t$ has four real roots, the expression $r_1(t)+r_4(t)-r_2(t)-r_3(t)$ has a constant sign. We can observe that $\displaystyle \sum_{1\le i\le 4} r_i(t),\ \sum_{1\le i<j\le 4}r_i(t)r_j(t),\ \sum_{1\le i<j<k\le 4}r_i(t)r_j(t)r_k(t)$ do not depend on $t$, as they depend only on the first $4$ coefficients of $P_t$, which are the same for all $t$. This means that $(r_1+r_2-r_3-r_4)(r_1+r_3-r_2-r_4)(r_1+r_4-r_2-r_3)$ is constant for all $t$ (I wrote $r_i$ instead of $r_i(t)$), and from the fact that $r_1+r_2-r_3-r_4<0,\ r_1+r_3-r_2-r_4<0$, we derive what we want: $r_1+r_4-r_2-r_3$ has a constant sign for all $t$ for which $P_t$ has four real roots.