Polynomial Functions Problem

Question

If $f(x)=x^2+1$ and $g(x)=x-1$, for all real numbers $x$, for what real number $a$ does $f(g(-a))= g(f(-a))$?

Answer 1

Given $f(x)=x^2+1$ and $g(x)=x-1$

$$f(g(-a))=g(f(-a))$$ $$f(-a-1)=g(a^2+1)$$ $$a^2+2+2a=a^2+1-1$$ $$a=-1$$

Answer 2

Write $$f(g(-x)) = (-(x-1))^2+1$$ so, expand it. Do the same for $g(f(-x))$.

Thus, you get a equality of the form $$ax^2+bx+c=px^2+qx+r.$$

Just solve it.