Hey, I'm a third year Maths student and I also work as a tutor for first and second years at my university. Just about that complex number problem, particularly in regards to the whole x=/=0 thing:
Let w = u +iv (i.e. real and imaginary part), but Im(w) = 0 so w = u (where u is real)
From the line w(z^2 + 1) = z:
u(x^2 - y^2 +1 +2ixy) = x +iy
Equate real and imaginary:
x = u(x^2 - y^2 +1) and y = 2xyu
Look at the second of these simultaneous equations first: y = 2xyu => y - 2xyu = 0 => y(1 - 2xu) = 0.
Since we are given y =/= 0 we have (1 - 2xu) = 0. Looking at this equation, if x = 0 then we would have 1 = 0, which means that there is no solution with x = 0 for these two simultaneous equations. So we're not just assuming x=/=0, we've arrived at the conclusion that x=/=0 through our working.
We can then put u=1/(2x) and substitute that into the first equation, which with a little bit of working leads to x^2 +y^2 = 1
From there it should be clear to anyone doing a class involving complex numbers that |z| = 1. There is essentially no step in working between those two lines, you just need to know the definition of the modulus of a complex number as thesoaf said.
Note: Just looking at the given information of the problem, look what happens when we put x=0, so z = iy (purely imaginary)
(We have that y=/=1 here, because we are given (z^2+1)=/=0 which if x=0 => y=/=1)
w=z/(z^2 + 1) would give w = yi / (1 - y^2) = i (y/(1-y^2))
This is a purely imaginary number. But we are given that Im(w) = 0, which would mean that y=0. But we are also given y=/=0, so assuming x=0 and arriving at y=0 is proof by contraciction that x=/=0.
So it's never just assumed that x=/=0. Even though we aren't explicitly given that in the problem, x=/=0 is implied by the two bits of information we are given, namely that y=/=0 and Im(w)=0.