Given inequality -: x2 + 2xy + 2y2 > 0 always satisfies the inequality for every x and y, where x and y are real numbers(x and y are not equal to 0).
How to prove this without using any graph? Is there any mathematical formula or something like that?
Second example -: x2 + 10xy + 2y2 > 0 does not satisfy the inequality always.
It is $$x^2+2xy+y^2+y^2=(x+y)^2+y^2>0$$