1. **Problem statement:** Find the products of the expressions: i) $ (a - 1)(a^2 + a + 1) $ ii) $ (8 + b)(64 - 8b + b^2) $ 2. **Solution for i):** Use distributive property (FOIL for polynomials): $$ (a - 1)(a^2 + a + 1) = a(a^2 + a + 1) - 1(a^2 + a + 1) $$ $$ = a^3 + a^2 + a - a^2 - a - 1 $$ Simplify by combining like terms: $$ a^3 + (a^2 - a^2) + (a - a) - 1 = a^3 - 1 $$ 3. **Solution for ii):** Recognize the pattern as a product of sum and difference of cubes: $$ (8 + b)(64 - 8b + b^2) $$ Note that $8 = 2^3$ and $64 = 4^3$ but here better to multiply directly: Multiply each term: $$ 8 imes 64 = 512 $$ $$ 8 imes (-8b) = -64b $$ $$ 8 imes b^2 = 8b^2 $$ $$ b imes 64 = 64b $$ $$ b imes (-8b) = -8b^2 $$ $$ b imes b^2 = b^3 $$ Sum all terms: $$ 512 - 64b + 8b^2 + 64b - 8b^2 + b^3 $$ Combine like terms: $$ 512 + (-64b + 64b) + (8b^2 - 8b^2) + b^3 = 512 + 0 + 0 + b^3 = b^3 + 512 $$ 4. **Final answers:** i) $a^3 - 1$ ii) $b^3 + 512$