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

Synthetic division is a shortcut method for dividing polynomials that makes the process efficient and straightforward. When dividing \(x^4 - 18x^3 - 39x^2 + 17x + 30\) by \(x + 2\), we first take the root of the divisor, which is \(-2\). We will use the coefficients of the dividend: \(1, -18, -39, 17, 30\). Setting up synthetic division: ``` -2 | 1 -18 -39 17 30 | -2 40 -2 -30 ----------------------------- 1 -20 1 15 0 ``` The bottom row shows the coefficients of the quotient polynomial: \(1x^3 - 20x^2 + 1x + 15\) with a remainder of \(0\). Thus, the result is the polynomial \(x^3 - 20x^2 + x + 15\). To confirm this is correct, simply multiply back the quotient by the divisor and ensure it yields the original polynomial. It’s a great way to practice and verify your polynomial division skills!