Question Use synthetic division to find the result when \( 4 x^{4}+2 x^{3}-2 x^{2}+23 x-2 \) is divided by \( x+2 \). If there is a remainder, express the result in the form \( q(x)+\frac{r(x)}{b(x)} \).

To divide \( 4x^{4} + 2x^{3} - 2x^{2} + 23x - 2 \) by \( x + 2 \) using synthetic division, follow these steps: 1. **Identify the Root:** The divisor is \( x + 2 \), so the root is \( -2 \). 2. **Set Up the Synthetic Division:** Write down the coefficients of the polynomial: \[ 4 \quad 2 \quad -2 \quad 23 \quad -2 \] 3. **Perform the Division:** \[ \begin{array}{c|ccccc} -2 & 4 & 2 & -2 & 23 & -2 \\ & & -8 & 12 & -20 & 6 \\ \hline & 4 & -6 & 10 & 3 & -8 \\ \end{array} \] - **Step-by-Step Calculations:** - Bring down the **4**. - Multiply **4** by **-2** to get **-8**. Add this to **2** to get **-6**. - Multiply **-6** by **-2** to get **12**. Add this to **-2** to get **10**. - Multiply **10** by **-2** to get **-20**. Add this to **23** to get **3**. - Multiply **3** by **-2** to get **-6**. Add this to **-2** to get a **remainder** of **-8**. 4. **Write the Result:** Since there is a remainder, the division result is expressed as: \[ 4x^{3} - 6x^{2} + 10x + 3 - \frac{8}{x + 2} \] **Final Answer:** \[ 4x^{3} - 6x^{2} + 10x + 3 - \frac{8}{\,x + 2\,} \]