(a²-1)(a³-1)

To simplify the expression \((a^2 - 1)(a^3 - 1)\), we can use the factorizations of \(a^2 - 1\) and \(a^3 - 1\). 1. **Factor \(a^2 - 1\)**: \[ a^2 - 1 = (a - 1)(a + 1) \] 2. **Factor \(a^3 - 1\)**: \[ a^3 - 1 = (a - 1)(a^2 + a + 1) \] Now, we can substitute these factorizations back into the original expression: \[ (a^2 - 1)(a^3 - 1) = (a - 1)(a + 1)(a - 1)(a^2 + a + 1) \] Next, we can combine the common factors: \[ = (a - 1)^2(a + 1)(a^2 + a + 1) \] Thus, the simplified expression is: \[ \boxed{(a - 1)^2(a + 1)(a^2 + a + 1)} \]