Consider $$ \Phi(x)=\sum_{i=1}^n a_i^2 (1+x_i)^2\sum_{j=1}^n a_j^2 (1+x_j)(x_j-x_i)(x_1-x_i(2+x_j)),\quad a_i\in\mathbb R. $$ My conjecture is that this functional is non-negative for all $n$ and $x_i\geq 0$. Is it true?
My approach is to put $x=t y$, where $y\in\mathbb R^n$ is a unit non-negative vector and then rewrite $\Phi$ as a 6th-degree polynomial of $t$. Then I try to show that all its coefficients are positive (which is of course not a necessary condition). So far I think I have shown that the coefficients of $t^6$ and $t^5$ are always positive (using induction on $n$). The other coefficients are a bit clumsier and I wonder if there's a simpler way to prove the conjecture.
Random counterexample: $\Phi(x)=-288$ when $n=3,\,(a_1,a_2,a_3)=(0,4,3)$ and $x=(5,0,1)$.