u/Alternative-Cup-2527

Over Z_2, it factors as x^2(x+1) + (x+1) = (x^2 + 1)(x + 1) = (x^2 - 1)(x + 1) = (x-1)(x+1)^2 = (x+1)^3. But my question is, because we are only dealing with integers and an integer always has the same parity (odd/even) as its cube, doesn't (x+1)^3 reduce to x+1 over Z_2? AI keeps saying no but why not? Or am I right?