Use synthetic division to divide. \[ \frac{7 x^{3}+24 x^{2}+6 x-9}{x+3} \]

To perform synthetic division of \( 7x^3 + 24x^2 + 6x - 9 \) by \( x + 3 \), we first set \( x + 3 = 0 \) to find the value we will use in synthetic division, which gives us \( x = -3 \). Now, we organize the coefficients of the polynomial: \( 7, 24, 6, -9 \). Here’s how we do synthetic division: 1. Write down the coefficients: \( 7, 24, 6, -9 \) 2. Write \( -3 \) to the left: \[ \begin{array}{r|rrrr} -3 & 7 & 24 & 6 & -9 \\ \end{array} \] 3. Bring down the first coefficient (7): \[ \begin{array}{r|rrrr} -3 & 7 & 24 & 6 & -9 \\ & & & & \\ & 7 & & & \\ \end{array} \] 4. Multiply \( 7 \) by \( -3 \) and place it under the next coefficient (24): \[ \begin{array}{r|rrrr} -3 & 7 & 24 & 6 & -9 \\ & & -21 & & \\ & 7 & & & \\ \end{array} \] 5. Add \( 24 + (-21) = 3 \): \[ \begin{array}{r|rrrr} -3 & 7 & 24 & 6 & -9 \\ & & -21 & -9 & \\ & 7 & 3 & & \\ \end{array} \] 6. Multiply \( 3 \) by \( -3 \) and place it under the next coefficient (6): \[ \begin{array}{r|rrrr} -3 & 7 & 24 & 6 & -9 \\ & & -21 & -9 & \\ & 7 & 3 & & \\ \end{array} + \begin{array}{r|rrrr} & & & -9 & \\ \end{array} \] 7. Add \( 6 + (-9) = -3 \): \[ \begin{array}{r|rrrr} -3 & 7 & 24 & 6 & -9 \\ & & -21 & -9 & \\ & 7 & 3 & -3 & \\ \end{array} \] 8. Multiply \( -3 \) by \( -3 \) and place it under the last coefficient (-9): \[ \begin{array}{r|rrrr} -3 & 7 & 24 & 6 & -9 \\ & & -21 & -9 & 9 \\ & 7 & 3 & -3 & \\ \end{array} \] 9. Add \( -9 + 9 = 0 \). Now, we write the result from top to bottom. The new polynomial is: \[ 7x^2 + 3x - 3 \] with a remainder of 0. Thus, the final result of the synthetic division is: \[ \frac{7x^3 + 24x^2 + 6x - 9}{x + 3} = 7x^2 + 3x - 3 \]