Explanation

Write x^6=(x^2)^3. Dividing y^3 by y+1 (y=x^2): y^3=(y+1)(y^2-y+1)-1, since (y+1)(y^2-y+1)=y^3+1. The remainder is -1. Equivalently, x^2=-1 (mod x^2+1) gives x^6=(-1)^3=-1.