1. **Problem:** Given that $9x^{2} + 6xy + 4y^{2}$ is a factor of $27x^{3} - 8y^{3}$, find the other factor. 2. **Step 1: Recognize the form of the expression.** The expression $27x^{3} - 8y^{3}$ is a difference of cubes since $27x^{3} = (3x)^{3}$ and $8y^{3} = (2y)^{3}$. 3. **Step 2: Recall the factorization formula for difference of cubes:** $$a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})$$ where $a = 3x$ and $b = 2y$. 4. **Step 3: Apply the formula:** $$27x^{3} - 8y^{3} = (3x - 2y)((3x)^{2} + (3x)(2y) + (2y)^{2})$$ $$= (3x - 2y)(9x^{2} + 6xy + 4y^{2})$$ 5. **Step 4: Identify the other factor.** Since $9x^{2} + 6xy + 4y^{2}$ is given as one factor, the other factor must be $3x - 2y$. 6. **Step 5: Match with the options given:** From the second set of options: A. $3x - 2y$ B. $2y - 3x$ C. $2y + 3x$ D. $3x + 2y$ The correct other factor is option A: $3x - 2y$. **Final answer:** The other factor is $3x - 2y$.