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

To find the quotient and remainder when dividing \( \frac{x^{3} + 3x + 2}{x^{2} - x + 4} \) using long division, we proceed as follows: 1. Divide the leading term of the numerator \( x^3 \) by the leading term of the denominator \( x^2 \) to get \( x \). 2. Multiply the entire divisor \( x^2 - x + 4 \) by \( x \) resulting in \( x^3 - x^2 + 4x \). 3. Subtract this result from the original numerator: \[ (x^3 + 3x + 2) - (x^3 - x^2 + 4x) = x^2 - x + 2. \] 4. Now, repeat the process by dividing the leading term \( x^2 \) by \( x^2 \) to get \( 1 \). 5. Multiply the divisor by \( 1 \): \( x^2 - x + 4 \). 6. Subtract this from the current polynomial: \[ (x^2 - x + 2) - (x^2 - x + 4) = -2. \] At this point, the degree of the resulting polynomial (-2) is lower than the degree of the divisor (2), so we stop. The final answer is: - Quotient: \( x + 1 \) - Remainder: \( -2 \) Thus, - Quotient: \( x + 1 \) - Remainder: \( -2 \)