|-8z+4| = 20

z = -2

Again the book's anwser is -2, 3

Apparently I am missing a step. What is it?

It's because of the absolute brackets.

The absolute brackets | | mean that it is either 20 or -20 so using the -20 and solving for z would give you the book's answer of 3 but using the 20 would give you -2 for z