Prove the inequality $\sum_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} \ge 0 $

Question

A is a square matrix with positive elements and x is a real vector (both of them n>1 dimensional). Prove that for any such matrix and vector $$\sum\limits_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} \ge 0 $$ P.S.#1: I am not 100% sure that the inequality holds but I have seen it happen in numerous numerical examples (for random matrices and vectors), P.S.#2: I need this to prove the uniqueness of the solution to a PDE.