First thing to note is that it is a quadratic, so you will get 2 solutions (roots) to the equation.
You are correct to factorize the z which is common in each term i.e.
z(z+2+i) = 0
This implies that z = 0 is the first solution.
Then you have z+2+i=0 which when re-arranged gives
z = -2-i
You can check that you are correct by substituting in these values for z into the original quadratic equation - you will find the left hand side equates to 0 for both z's as required.