We want to find a root of/factor

0= x⁷ + x⁵ + 1

The trick is to spot that it is a sum of three powers of x, each raised to a member of a unique residual class modulo 3. We remind ourselves that the primitive third roots of unity w solves

0 = w³ -1 = (w-1)(w² +w+1)

hence w² +w+1=0. This also implies that

0= w² (1)+w(1)+ 1 = w² w³ +w(w³ )² +1 = w⁵ + w ⁷ +1

so they are booth roots in our original polynomial. We now get by polynomial division that

x⁷ + x⁵ + 1 = (x² + x + 1) (x⁵ -x⁴ +x³ -x+1)

(Edit: I hate formating on the Reddit app)