<s>[INST] show that the function $z = 2x^2 + y^2 +2xy -2x +2y +2$ is greater than $-3$ Show that the function 
$$z = 2x^2 + y^2 +2xy -2x +2y +2$$
is greater than $-3$
I tried to factorize but couldn't get more than $(x-1)^2 + (x+y)^2 +(y-1)^2 - (y)^2$.
Is there any another way to factorize or another method??
 [/INST] HINT:
$$z = 2x^2 + y^2 +2xy -2x +2y +2=(x+y+1)^2+(x-2)^2+2-4-1$$
Derivation :
Let $$ 2x^2 + y^2 +2xy -2x +2y +2=(x+y+a)^2+(x+b)^2+2-a^2-b^2$$
$$\implies  2x^2 + y^2 +2xy -2x +2y +2=2x^2+y^2+2xy+2x(a+b)+2ay+2-a^2-b^2$$
Comparing the coefficients of $y, a=1$
Comparing the coefficients of $x, a+b=-1\implies b=-1-a=-2$
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