1. **Problem 1:** Factorize the cubic polynomial $x^3 - 13x^2 + 24x + 108$ given that one factor is $(x + 2)$. We want to find the other two factors. 2. Use polynomial division or synthetic division to divide $x^3 - 13x^2 + 24x + 108$ by $(x + 2)$. 3. Dividing, we get quotient $x^2 - 15x + 54$. 4. Now factor $x^2 - 15x + 54$. Find two numbers that multiply to $54$ and add to $-15$: these are $-6$ and $-9$. 5. So, $x^2 - 15x + 54 = (x - 6)(x - 9)$. 6. Therefore, the full factorization is: $$x^3 - 13x^2 + 24x + 108 = (x + 2)(x - 6)(x - 9)$$ --- 1. **Problem 2:** Perform polynomial division of $x^4 + 3x^3 + 2x^2 - 9x + 6$ by $x + 3$. 2. Use long division: - Divide $x^4$ by $x$ to get $x^3$. - Multiply $(x + 3)(x^3) = x^4 + 3x^3$. - Subtract: $(x^4 + 3x^3) - (x^4 + 3x^3) = 0$. - Bring down $2x^2$. - Divide $2x^2$ by $x$ to get $2x$. - Multiply $(x + 3)(2x) = 2x^2 + 6x$. - Subtract: $(2x^2 - 9x) - (2x^2 + 6x) = -15x$. - Bring down $+6$. - Divide $-15x$ by $x$ to get $-15$. - Multiply $(x + 3)(-15) = -15x - 45$. - Subtract: $(-15x + 6) - (-15x - 45) = 51$. 3. The quotient is $x^3 + 2x - 15$ with remainder $51$. 4. So, $$\frac{x^4 + 3x^3 + 2x^2 - 9x + 6}{x + 3} = x^3 + 2x - 15 + \frac{51}{x + 3}$$ --- **Final answers:** - Factorization: $(x + 2)(x - 6)(x - 9)$ - Polynomial division quotient: $x^3 + 2x - 15$ with remainder $51$