please explain the circled steps.

smple quadratic equation solving.....

i can understand mod z² but how mod z is coming?

That's because
\dfrac{a^2+2-\sqrt{a^4+4a^2}}{2}
=\dfrac{2a^2+4-2\sqrt{a^2(a^2+4)}}{4}
=\dfrac{a^2+(a^2+4)-2a\sqrt{(a^2+4)}}{4}
=\left(\dfrac{\sqrt{a^2+4}-a}{2}\right)^2
And similarly for the other one.

why not?

( (a-√a²+4)²/2)²