Families of polynomial question?

Question

Given a function g(x) such that g(x^2+2)=x^4+5x^2+3, determine g(x^2-1).

The answer is g(x^2-1) = x^4 -x^2 -3

How do I get this?

Answer 1

Given,  g(x²+2)=x⁴+5x²+3 = (x² +2)² + (x² +2) -3

Let y = x²+2,, then g(y) = y² + y -3

This is a function of y, so if y= x² -1, then 

g(x² -1) =  (x² -1)² + (x² -1) - 3

             =  x⁴ - 2x² +1 + x² -1 -3

             = x⁴ - x² -3