1. **State the problem:** Simplify the expression $ab^2 - a^3$. 2. **Identify common factors:** Both terms have a common factor of $a$. 3. **Factor out the common factor:** $$ab^2 - a^3 = a(b^2) - a(a^2) = a(b^2 - a^2)$$ 4. **Recognize the difference of squares:** $$b^2 - a^2 = (b - a)(b + a)$$ 5. **Write the fully factored form:** $$a(b - a)(b + a)$$ 6. **Explanation:** We factored out the common factor $a$ first, then used the difference of squares formula to factor $b^2 - a^2$. This is the simplest factorization of the expression.