Composite Function Exam Question - Unknown Function - Hidden Quadratic - IB AA SL & HL Mathematics

We work through an exam question on composite functions.
We're given g(f(x)) as well as g(x) and we need to find two possible functions for f(x).
The solution involves working with a quadratic equation.
This question is based on a past IB AA SL & HL Mathematics Paper 1 question.