se the long division method to find the result when \( 3 x^{3}+23 x^{2}-x-10 \) is divided by \( x+2 \).

Sure! Let's use the **long division method** to divide the polynomial \( 3x^{3} + 23x^{2} - x - 10 \) by \( x + 2 \). ### Step-by-Step Long Division 1. **Set Up the Division** \[ \begin{array}{r|lll} & 3x^3 & +23x^2 & -x & -10 \\ x + 2 \, ) & \underline{3x^3} & +23x^2 & -x & -10 \\ \end{array} \] 2. **Divide the First Terms** - **Divide:** \( 3x^3 \div x = 3x^2 \) - **Write:** \( 3x^2 \) above the division bar. \[ \begin{array}{r|llll} & 3x^2 & & & \\ x + 2 \, ) & 3x^3 & +23x^2 & -x & -10 \\ & \underline{3x^3} & +6x^2 & & \\ & & 17x^2 & -x & \\ \end{array} \] 3. **Multiply and Subtract** - **Multiply:** \( 3x^2 \times (x + 2) = 3x^3 + 6x^2 \) - **Subtract:** \[ (3x^3 + 23x^2) - (3x^3 + 6x^2) = 17x^2 \] - **Bring Down:** The next term \(-x\). 4. **Repeat the Process** - **Divide:** \( 17x^2 \div x = 17x \) - **Write:** \( +17x \) above the division bar. \[ \begin{array}{r|llll} & 3x^2 & +17x & & \\ x + 2 \, ) & 3x^3 & +23x^2 & -x & -10 \\ & \underline{3x^3} & +6x^2 & & \\ & & 17x^2 & -x & \\ & & \underline{17x^2} & +34x & \\ & & & -35x & \\ \end{array} \] 5. **Multiply and Subtract Again** - **Multiply:** \( 17x \times (x + 2) = 17x^2 + 34x \) - **Subtract:** \[ (17x^2 - x) - (17x^2 + 34x) = -35x \] - **Bring Down:** The next term \(-10\). 6. **Final Division Step** - **Divide:** \( -35x \div x = -35 \) - **Write:** \( -35 \) above the division bar. \[ \begin{array}{r|llll} & 3x^2 & +17x & -35 & \\ x + 2 \, ) & 3x^3 & +23x^2 & -x & -10 \\ & \underline{3x^3} & +6x^2 & & \\ & & 17x^2 & -x & \\ & & \underline{17x^2} & +34x & \\ & & & -35x & -10 \\ & & & \underline{-35x} & -70 \\ & & & & 60 \\ \end{array} \] 7. **Multiply and Subtract One Last Time** - **Multiply:** \( -35 \times (x + 2) = -35x - 70 \) - **Subtract:** \[ (-35x - 10) - (-35x - 70) = 60 \] ### Final Result - **Quotient:** \( 3x^2 + 17x - 35 \) - **Remainder:** \( 60 \) So, the division can be expressed as: \[ \frac{3x^{3} + 23x^{2} - x - 10}{x + 2} = 3x^2 + 17x - 35 + \frac{60}{x + 2} \] **Summary:** \[ 3x^{3} + 23x^{2} - x - 10 = (x + 2)(3x^2 + 17x - 35) + 60 \]