Prove polynomial inequality

Question

I need to prove that a function f has a minimum, among convexity analysis I need to prove that

$f(x,y)≥2(x^2+y^2 )-2 \sqrt{(x^2+y^2)}-8$

with

$f(x,y)=x^4+y^4-2x^2+x+y-3$

EDIT

$(x,y) \in \mathbb{R}^2$

Answer 1

To prove the result, I just had to start with the following inequality

$x+y > -2\sqrt{(x^2+y^2)}$

and add polynomial terms in canonical form on the LHS.