Verified answer

Note that x^(p-2) + x^(p-1) + ... + x^2 + x + 1

= (x^(p-1) - 1)/(x - 1).

For all x ≠ 1 (mod p), we see that (x - 1)^(-1) exists mod p.

Therefore, since x^(p-1) = 1 (mod p) for all x = 2, 3, ..., p-1 via Fermat's Little Theorem,

we conclude that x^(p-2) + x^(p-1) + ... + x^2 + x + 1 = 0 (mod p).

Since x = 1 (mod p) is clearly not a solution of x^(p-2) + x^(p-1) + ... + x^2 + x + 1 = 0 (mod p),

we conclude that this congruence has exactly (p-2) incongruent solutions.

I hope this helps!