Find all $z$ for which $z^2+2z+2$ is real positive. Find all $z$ for which $z^2+2z+2$ is real positive. 
My Attempt :
Let $z=x+iy,  (x,y\in R)$:
$$z^2+2z+2= (x^2-y^2+2ixy) +2(x+iy)+2$$
$$=(x^2-y^2+2x+2)+i(2xy+2y)$$

You want the real part to be positive and the imaginary part to be $0$
$$ (x^2-y^2+2x+2)+i(2xy+2y)$$
$$2xy+2y=0 \implies 2y(x+1)=0$$
Thus  $y=0$ or $x=-1$
For $y=0$ we get $x^2+2x+2=(x+1)^2+1 >0$
For $ x=-1 $ we need $1-y^2>0$ which implies $|y|<1$
Thus the solution is either $ y=0$ or ($|y|<1$ and $x=-1$ )    
