1. **State the problem:** Find the Least Common Multiple (LCM) of the algebraic expressions $10 + 10x$ and $(2x + 2)(5x - 10)$. 2. **Rewrite expressions to factor form:** - $10 + 10x = 10(1 + x)$ - $2x + 2 = 2(x + 1)$ - $5x - 10 = 5(x - 2)$ So, $(2x + 2)(5x - 10) = 2(x + 1) \times 5(x - 2) = 10(x + 1)(x - 2)$. 3. **Identify common and unique factors:** - First expression factors: $10(1 + x)$ - Second expression factors: $10(x + 1)(x - 2)$ Note that $1 + x$ and $x + 1$ are equivalent. 4. **LCM rule:** The LCM of algebraic expressions is the product of the highest powers of all factors appearing in any expression. 5. **Combine factors for LCM:** - Both have factor $10$ - Both have factor $(x + 1)$ - Only second has factor $(x - 2)$ Therefore, $$\text{LCM} = 10(x + 1)(x - 2)$$ 6. **Final answer:** The LCM of $10 + 10x$ and $(2x + 2)(5x - 10)$ is $$10(x + 1)(x - 2)$$.