Better method to show $-x^3 + 1 = (-x + 1)(x^2 + x + 1)$ Is there a rule, algorithm, or theorem used for this equality below:
$$-x^3 + 1 = (-x + 1)(x^2 + x + 1)$$
I know the RHS can be established using trial and error distributing the first term through the second, or alternatively same with FOIL.  I'm curious if there is a better method?
Image below from a CAS.


Let $\zeta$ be a primitive cube root of unity, then $\zeta^3=1,\zeta^2+\zeta+1=0$ and hence $1-x^3=-(x^3-1)=-(x-1)(x-\zeta)(x-\zeta^2)=(1-x)(x^2-(\zeta+\zeta^2)x+\zeta^3)=(1-x)(x^2+x+1)$
Note: FOIL, which is the distributive property of multiplication and addition, is unavoidable in pretty much any proof. To try to avoid using FOIL in a proof would be like trying to avoid using the fact that $1+1=2\ne 0$ in a field of characteristic $\gt 2$ or $0$.
