1. The problem involves simplifying and factoring cubic expressions such as $100e^3 + 400 \, 40e + 16$, $125 + 216b^3$, and $429a^3 - 27b^3$. 2. Recall the sum and difference of cubes formulas: - Sum of cubes: $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ - Difference of cubes: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ 3. For the expression $125 + 216b^3$, recognize $125 = 5^3$ and $216b^3 = (6b)^3$. Using the sum of cubes formula: $$125 + 216b^3 = 5^3 + (6b)^3 = (5 + 6b)(25 - 30b + 36b^2)$$ 4. For $429a^3 - 27b^3$, note $27b^3 = (3b)^3$. Factor using the difference of cubes: $$429a^3 - 27b^3 = (\sqrt[3]{429}a - 3b)(\text{quadratic expression})$$ However, since 429 is not a perfect cube, this expression may require factoring out common factors or alternative methods. 5. The expression $81 - 64b^3$ can be seen as $3^4 - (4b)^3$, which is not a direct cube sum or difference but can be analyzed further for factorization. 6. The first expression $100e^3 + 400 \, 40e + 16$ appears incomplete or mistyped; if it is $100e^3 + 400e + 16$, it may require grouping or other factoring techniques. Summary: - Use sum and difference of cubes formulas to factor expressions like $125 + 216b^3$. - For expressions involving cubes but not perfect cubes, consider factoring common terms or alternative methods. Final factored form for $125 + 216b^3$ is: $$ (5 + 6b)(25 - 30b + 36b^2) $$