\[ \begin{array}{l}\frac{x^{3}+3 x+2}{x^{2}-x+4} \\ \text { Find the quotient and remainder using long division. } \\ \text { quotient } \\ \text { remainder } \\ \square\end{array} \]

To find the quotient and remainder of the polynomial division \(\frac{x^{3}+3x+2}{x^{2}-x+4}\), we apply long division: 1. Divide the leading term of the numerator \(x^3\) by the leading term of the denominator \(x^2\), which gives \(x\). 2. Multiply the entire denominator \(x^2 - x + 4\) by \(x\): \(x(x^2 - x + 4) = x^3 - x^2 + 4x\). 3. Subtract this from the original numerator: \((x^3 + 3x + 2) - (x^3 - x^2 + 4x) = x^2 - x + 2\). 4. Now, repeat the steps: Divide the leading term \(x^2\) by \(x^2\), giving \(1\). 5. Multiply the denominator by \(1\): \(1(x^2 - x + 4) = x^2 - x + 4\). 6. Subtract again: \((x^2 - x + 2) - (x^2 - x + 4) = -2\). Thus, the division results in: - **Quotient**: \(x + 1\) - **Remainder**: \(-2\) The final answer is: - Quotient: \(x + 1\) - Remainder: \(-2\)