QUESTION 25 If possible, factor the polynomial completely. If a polynomial cannot be factored, state that it is prime. \( 9 a^{2}+30 a+25 \)

To factor the polynomial \( 9a^{2} + 30a + 25 \) completely, let's follow these steps: 1. **Identify the Structure:** The given polynomial is a quadratic in the form \( ax^2 + bx + c \). 2. **Look for Factoring Patterns:** We can attempt to factor it as \( (ma + n)(pa + q) \). 3. **Determine the Coefficients:** - The product of the first terms should be \( 9a^2 \). Possible pairs are: - \( 3a \times 3a = 9a^2 \) - The product of the last terms should be \( 25 \). Possible pairs are: - \( 5 \times 5 = 25 \) 4. **Verify the Middle Term:** Now, check if the middle terms add up to \( 30a \): \[ (3a \times 5) + (3a \times 5) = 15a + 15a = 30a \] This matches the middle term of the original polynomial. 5. **Write the Factored Form:** Thus, the polynomial factors neatly into: \[ (3a + 5)^2 \] **Final Answer:** The polynomial factors completely as (3 a + 5) squared. **Answer:** The polynomial factors completely as (3 a + 5) squared.