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 $2xy+2y = 2y(x+1)=0$ and $x^2-y^2+2x+2 = (x+1)^2-y^2+1>0$. From the first equation, $y=0$ or $x=-1$; if $y=0$, you want $(x+1)^2+1>0$, which is always true; if $x=-1$, you want $-y^2+1>0$, which is true when $|y|<1$. So the solutions are 
\begin{equation*}
  \{x+iy\,|\,y=0\text{ or }(x=-1\text{ and }|y|<1)\}.
\end{equation*}
