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The given polynomial is \( 9x^3 + 6x^2 + 4x + 2 \), and the question asks which of the provided options is a factor of this polynomial.

We will attempt to factor this expression by checking the potential factors given in the options.

### Step-by-step factorization process:

#### 1. Try \( 3x + 1 \):
Use synthetic division to divide \( 9x^3 + 6x^2 + 4x + 2 \) by \( 3x + 1 \).

#### 2. Try \( x - 3 \):
Use synthetic or polynomial division to check if \( x - 3 \) is a factor.

#### 3. Try \( 2x + 3 \):
Similarly, we can check if \( 2x + 3 \) is a factor.

We can systematically divide to check which, if any, of these expressions divides the polynomial without a remainder.

Let me calculate this for you.None of the divisions produced a simplified result, meaning that none of the given options (\(3x + 1\), \(x - 3\), or \(2x + 3\)) are exact factors of the polynomial \(9x^3 + 6x^2 + 4x + 2\).

Thus, the correct answer is: **Not factorable**.

Would you like more details on the division process or have any questions? Here are 5 related questions:

1. How does synthetic division work for polynomials?
2. Can we use the Rational Root Theorem to check for factors?
3. How would you factor a cubic polynomial if it were factorable?
4. What is the significance of remainder in polynomial division?
5. How can polynomial long division help in factorization?

**Tip:** Always check for common factors before performing polynomial division to simplify the process!

Which of the following is a factor of 9x³ + 6x² + 4x + 2?

Explore the factorization of the cubic polynomial 9x³ + 6x² + 4x + 2. Understand the use of synthetic division and the Factor Theorem in determining factors. Suitable for students in Grades 10-12.

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