You have the following algebraic expression:
x³ - 1
In order to determine which of the given factor corresponds to the given expression, it is necessary that the division between x³ + 1 and x -1 or x² -1 has zero remainder.
Then, you calculate the division, just as follow:
x³ - 1 | x - 1
-x³ +x² x² + x + 1
x² -1
-x²+ x
x - 1
-x + 1
0
As you can notice, the division between x³ - 1 and x - 1 has zero remainder.
The multiplication between the divisor and the quotient is the factorized form of the given expression, then, you have:
x³ - 1 = (x - 1)(x² + x + 1)