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 $z=x+iy,  (x,y\in R)$:
$$z^2+2z+2= (x^2-y^2+2ixy) +2(x+iy)+2$$
$$=(x^2-y^2+2x+2)+i(2xy+2y)$$

Let $p$ be a real number.
$$(z+1)^2+1=p^2\iff z=\pm\sqrt{p^2-1}-1.$$
If $|p|<1$, $z=\pm i\sqrt{1-p^2}-1$ runs from $-1-i$ to $-1+i$,
If $|p|\ge1$, $z=\pm\sqrt{p^2-1}-1$, runs from $-\infty$ to $\infty$.
