\( \left. \begin{array} { l } { m ^ { 3 } + 27 } \\ { ( m + 3 ) ^ { 3 } } \\ { ( m + 3 ) ( m ^ { 2 } + 9 ) } \\ { ( m - 3 ) ( m ^ { 2 } + 3 m + 9 ) } \end{array} \right. \)

It looks like you've provided a set of algebraic expressions involving the variable \( m \). Let's analyze each expression to understand their relationships, especially focusing on polynomial factorization. ### 1. \( m^3 + 27 \) **Interpretation:** This is a **sum of cubes**, since \( 27 = 3^3 \). **Factorization:** The sum of cubes formula is: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] Applying this to \( m^3 + 27 \): \[ m^3 + 27 = m^3 + 3^3 = (m + 3)(m^2 - 3m + 9) \] ### 2. \( (m + 3)^3 \) **Interpretation:** This is the **expansion of a binomial raised to the third power**. **Expansion:** Using the binomial theorem or straightforward multiplication: \[ (m + 3)^3 = m^3 + 3 \cdot m^2 \cdot 3 + 3 \cdot m \cdot 3^2 + 3^3 = m^3 + 9m^2 + 27m + 27 \] **Comparison:** This is **not equal** to \( m^3 + 27 \) unless \( m = 0 \). ### 3. \( (m + 3)(m^2 + 9) \) **Interpretation:** This appears to be an attempted factorization, but it's **incomplete** for \( m^3 + 27 \). **Multiplication:** \[ (m + 3)(m^2 + 9) = m^3 + 9m + 3m^2 + 27 = m^3 + 3m^2 + 9m + 27 \] **Comparison:** This result does **not** match \( m^3 + 27 \) unless \( m = 0 \). Therefore, this is **not a correct factorization** of \( m^3 + 27 \). ### 4. \( (m - 3)(m^2 + 3m + 9) \) **Interpretation:** This is the factorization of a **difference of cubes**, specifically \( m^3 - 27 \). **Factorization:** The difference of cubes formula is: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] Applying this to \( m^3 - 27 \): \[ m^3 - 27 = m^3 - 3^3 = (m - 3)(m^2 + 3m + 9) \] ### **Summary of Correct Factorizations:** 1. **Sum of Cubes:** \[ m^3 + 27 = (m + 3)(m^2 - 3m + 9) \] 2. **Difference of Cubes:** \[ m^3 - 27 = (m - 3)(m^2 + 3m + 9) \] 3. **Expansion:** \[ (m + 3)^3 = m^3 + 9m^2 + 27m + 27 \] 4. **Incorrect Factorization Attempt:** \[ (m + 3)(m^2 + 9) \neq m^3 + 27 \] ### **Key Takeaways:** - **Sum of Cubes:** \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) - **Difference of Cubes:** \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \) - **Cubing a Binomial:** Expanding \( (a + b)^3 \) results in \( a^3 + 3a^2b + 3ab^2 + b^3 \), which is generally different from just \( a^3 + b^3 \). Ensure you use the correct formulas when factoring or expanding polynomials to avoid inconsistencies.