Divide using synthetic division. \( \begin{array}{l}\left(x^{4}-18 x^{3}-39 x^{2}+17 x+30\right) \div(x+2) \\ x^{3}+x^{2}+15 x-20 \\ x^{3}+22 x^{2}-41 x-13 \\ x^{3}-18 x^{2}+22 x-13 \\ x^{3}-20 x^{2}+x+15\end{array} \)

To perform synthetic division of the polynomial \( x^4 - 18x^3 - 39x^2 + 17x + 30 \) by \( x + 2 \), we first need to use the root of the divisor, which is \( -2 \). Here are the coefficients of the polynomial: - \( 1 \) (for \( x^4 \)) - \( -18 \) (for \( x^3 \)) - \( -39 \) (for \( x^2 \)) - \( 17 \) (for \( x^1 \)) - \( 30 \) (constant term) We will set up synthetic division as follows: 1. Write down the coefficients: \( 1, -18, -39, 17, 30 \). 2. Write \( -2 \) to the left. Now we perform the synthetic division: \[ \begin{array}{r|rrrrr} -2 & 1 & -18 & -39 & 17 & 30 \\ & & -2 & 40 & -2 & -30 \\ \hline & 1 & -20 & 1 & 15 & 0 \\ \end{array} \] ### Steps: - Bring down the leading coefficient \( 1 \). - Multiply \( 1 \) by \( -2 \) to get \( -2 \) and add it to \( -18 \) to get \( -20 \). - Multiply \( -20 \) by \( -2 \) to get \( 40 \) and add it to \( -39 \) to get \( 1 \). - Multiply \( 1 \) by \( -2 \) to get \( -2 \) and add it to \( 17 \) to get \( 15 \). - Multiply \( 15 \) by \( -2 \) to get \( -30 \) and add it to \( 30 \) to get \( 0 \). The final row gives us the coefficients of the quotient polynomial, which is \( x^3 - 20x^2 + 1x + 15 \), and the remainder is \( 0 \). Thus, the result of the division is: \[ \boxed{x^3 - 20x^2 + x + 15} \]