I must conclude something from this inequality, given that it must hold true for all values of $x_1$ and $x_2$. I believe it is ok to say $x_1$ and $x_2$ are the parameters and $a_{11}, a_{12},a_{21},a_{22}$ are the variables (the ones I want to find the relation between). The inequation is as follows:
$$x_1^2.a_{11} + x_2^2.a_{22} + x_1x_2.(a_{12} + a_{21}) \ge 0 ~~~\forall x_1, x_2.$$
So, I would like to know how to work with this kind of problem, what can I conclude from this inequality?
Your inequality can be written in matrix form as $(x_1,x_2)\left(\begin{array}{cc}a_{11}&a_{12}\\a_{21}& a_{22}\end{array}\right)\left(\begin{array}{c}x_1\\x_2\end{array}\right)\geq0$. This inequality is just telling you that the quadratic form associated to the matrix $\left(\begin{array}{cc}a_{11}&a_{12}\\a_{21}& a_{22}\end{array}\right)$ is positive semidefinite.