Short Answer 
The factor (x + 2b) is found only in Option D: 3x² + 39x + 18b after performing synthetic division on all options. Therefore, the correct polynomial is Option D.
Step 1: Select the Polynomials
We need to check four polynomials to see if (x + 2b) is a factor:
- Option A: 3x² + 9x + 18b
- Option B: 3x² + 24x + 18b
- Option C: 3x² + 30x + 18b
- Option D: 3x² + 39x + 18b
Step 2: Perform Synthetic Division for Each Polynomial
For each polynomial, substitute x = -2b to check if the output is zero, indicating that (x + 2b) is a factor. The conditions to check are:
- For Option A: 3(-2b)² + 9(-2b) + 18b = 12b² (not zero)
- For Option B: 3(-2b)¬≤ + 24(-2b) + 18b = 12b¬≤ Р30b (not zero)
- For Option C: 3(-2b)¬≤ + 30(-2b) + 18b = 12b¬≤ Р42b (not zero)
- For Option D: 3(-2b)¬≤ + 39(-2b) + 18b = 12b¬≤ Р60b (equals zero)
Step 3: Identify the Result
After checking all the polynomials, we determine that (x + 2b) is only a factor of Option D: 3x² + 39x + 18b. This means:
- The correct polynomial is Option D.
- Thus, the answer is 3.