We are given the polynomial function $f(x) = 2x^3 + x^2 - 18x - 9$ and the information that $(x + 3)$ is a factor. We are required to factorize the polynomial.

Since $(x + 3)$ is a factor, we can perform polynomial division to divide $f(x)$ by $(x + 3)$. Let's proceed with the division.

Step 1: Polynomial Division

We'll divide $2x^3 + x^2 - 18x - 9$ by $x + 3$.

1. Divide the first term: $\frac{2x^3}{x} = 2x^2$.
2. Multiply: $2x^2 \cdot (x + 3) = 2x^3 + 6x^2$.
3. Subtract: $(2x^3 + x^2) - (2x^3 + 6x^2) = -5x^2$.
4. Bring down the next term: $-5x^2 - 18x$.
5. Divide the first term: $\frac{-5x^2}{x} = -5x$.
6. Multiply: $-5x \cdot (x + 3) = -5x^2 - 15x$.
7. Subtract: $(-5x^2 - 18x) - (-5x^2 - 15x) = -3x$.
8. Bring down the next term: $-3x - 9$.
9. Divide the first term: $\frac{-3x}{x} = -3$.
10. Multiply: $-3 \cdot (x + 3) = -3x - 9$.
11. Subtract: $(-3x - 9) - (-3x - 9) = 0$.

The quotient is $2x^2 - 5x - 3$, and the remainder is 0, confirming that $(x + 3)$ is indeed a factor.

Step 2: Factorizing the Quotient

Now, we need to factor the quadratic $2x^2 - 5x - 3$.

To factorize $2x^2 - 5x - 3$, we can use the middle-term splitting method.

1. Find two numbers whose product is $2 \times -3 = -6$ and whose sum is $-5$.
   • The numbers are $-6$ and $1$ because $(-6) + 1 = -5$ and $(-6) \times 1 = -6$.
2. Rewrite the middle term: $2x^2 - 6x + x - 3$
3. Factor by grouping: $2x(x - 3) + 1(x - 3)$
4. Factor out the common factor: $(2x + 1)(x - 3)$

Final Factorization

The full factorization of $f(x) = 2x^3 + x^2 - 18x - 9$ is: $f(x) = (x + 3)(2x + 1)(x - 3)$

Would you like more details or have any questions?

Here are some related questions:

1. How do you determine the roots of the polynomial $f(x) = 2x^3 + x^2 - 18x - 9$ after factorization?
2. How can synthetic division be used to divide a polynomial instead of polynomial division?
3. What is the significance of the remainder being zero in polynomial division?
4. How can the factor theorem be applied to check if a binomial is a factor of a polynomial?
5. What other methods can be used to factorize cubic polynomials?

Tip: When factorizing a polynomial, always check if simple substitutions or factor theorem can be used before jumping into complex methods.