1. **State the problem:** We are given a cubic polynomial $f(x) = 3x^3 + 10x^2 + 9x + 2$ and told that $f(-2) = 0$. We need to determine the factors of $f(x)$. 2. **Recall the Factor Theorem:** If $f(a) = 0$ for some number $a$, then $(x - a)$ is a factor of $f(x)$. 3. **Apply the Factor Theorem:** Since $f(-2) = 0$, $(x + 2)$ is a factor of $f(x)$. 4. **Perform polynomial division:** Divide $f(x)$ by $(x + 2)$ to find the other factor. Set up the division: $$\frac{3x^3 + 10x^2 + 9x + 2}{x + 2}$$ - Divide the leading term: $3x^3 \div x = 3x^2$ - Multiply back: $3x^2(x + 2) = 3x^3 + 6x^2$ - Subtract: $(3x^3 + 10x^2) - (3x^3 + 6x^2) = 4x^2$ - Bring down $+9x$ - Divide: $4x^2 \div x = 4x$ - Multiply back: $4x(x + 2) = 4x^2 + 8x$ - Subtract: $(4x^2 + 9x) - (4x^2 + 8x) = x$ - Bring down $+2$ - Divide: $x \div x = 1$ - Multiply back: $1(x + 2) = x + 2$ - Subtract: $(x + 2) - (x + 2) = 0$ 5. **Write the quotient:** The quotient is $3x^2 + 4x + 1$. 6. **Factor the quadratic:** Factor $3x^2 + 4x + 1$. Find two numbers that multiply to $3 \times 1 = 3$ and add to $4$: these are $3$ and $1$. Rewrite: $$3x^2 + 3x + x + 1 = 3x(x + 1) + 1(x + 1) = (3x + 1)(x + 1)$$ 7. **Final factorization:** $$f(x) = (x + 2)(3x + 1)(x + 1)$$ **Answer:** The factors of $f(x)$ are $(x + 2)$, $(3x + 1)$, and $(x + 1)$.