Find partial fractions of $\frac{x^2-1}{x^4+1}$

Question

I tried to figure out how WA found the partial fractions of $\frac{x^2-1}{x^4+1}$:

Help is Appreciated.

Answer 1

It comes from $(x^2+x\sqrt{2}+1)(x^2-x\sqrt{2}+1) =(x^2+1)^2-(x\sqrt{2})^2 =x^4+2x^2+1-2x^2 =x^4+1 $ so that $\frac{x^2-1}{x^4+1} =\frac{x^2-1}{(x^2+x\sqrt{2}+1)(x^2-x\sqrt{2}+1)} $.

Then use partial fractions.