Use synthetic division to find the quotient and the remainder. \[ \left(x^{3}-3 x^{2}-9\right) \div(x+2) \]

To perform synthetic division of \( x^3 - 3x^2 - 9 \) by \( x + 2 \), we will use -2 (the opposite of 2) for the synthetic division process. 1. Write down the coefficients of the polynomial: \( 1 \) (for \( x^3 \)), \( -3 \) (for \( x^2 \)), \( 0 \) (for \( x \), since there is no \( x \) term), \( -9 \) (the constant term). 2. Set up the synthetic division: \[ \begin{array}{r|rrrr} -2 & 1 & -3 & 0 & -9 \\ & & -2 & 10 & -20 \\ \hline & 1 & -5 & 10 & -29 \\ \end{array} \] 3. Perform the synthetic division: - Bring down the \( 1 \). - Multiply \( 1 \) by \( -2 \) to get \( -2 \). Add this to \( -3 \) to get \( -5 \). - Multiply \( -5 \) by \( -2 \) to get \( 10 \). Add this to \( 0 \) to get \( 10 \). - Multiply \( 10 \) by \( -2 \) to get \( -20 \). Add this to \( -9 \) to get \( -29 \). The final row gives us the coefficients of the quotient and the remainder. Thus, the quotient is: \[ x^2 - 5x + 10 \] And the remainder is: \[ -29 \] So, we have: \[ \frac{x^3 - 3x^2 - 9}{x + 2} = x^2 - 5x + 10 \quad \text{remainder } -29. \]