Doubts with respect to a module maximum of a complex function.

Question

I want to find the maximum of $$ f (z) = (z + 1) ^ 2 $$ In the triangular region composed by the points $ (0,0), (2,0), (0, i) $ of the complex plane, I know that this maximum is on the edge of the triangle by a theorem, but I must analyze three paths, when analyzing the path in which x and y vary simultaneously, huge accounts are appearing when I replace $ y = - \frac {x} {2} +1 $ in the expression, is there any easier way to parse through this path? Is this substitution valid?

Answer 1

$|(z+1)^2|$ is the square of the distance from $-1$ to $z$. So, which point of your triangle is farthest from $-1$?