1. **State the problem:** Divide the polynomial $x^3 + x^2 - x + 2$ by $x + 4$ using long division. 2. **Recall the long division formula:** When dividing $P(x)$ by $D(x)$, we find quotient $Q(x)$ and remainder $R$ such that $$P(x) = D(x) \times Q(x) + R.$$ Here, $P(x) = x^3 + x^2 - x + 2$ and $D(x) = x + 4$. 3. **Perform the division step-by-step:** - Divide the leading term $x^3$ by $x$ to get $x^2$. - Multiply $x^2$ by $x + 4$ to get $x^3 + 4x^2$. - Subtract this from the original polynomial: $(x^3 + x^2) - (x^3 + 4x^2) = -3x^2$. - Bring down the next term $-x$ to get $-3x^2 - x$. - Divide $-3x^2$ by $x$ to get $-3x$. - Multiply $-3x$ by $x + 4$ to get $-3x^2 - 12x$. - Subtract: $(-3x^2 - x) - (-3x^2 - 12x) = 11x$. - Bring down the last term $+2$ to get $11x + 2$. - Divide $11x$ by $x$ to get $11$. - Multiply $11$ by $x + 4$ to get $11x + 44$. - Subtract: $(11x + 2) - (11x + 44) = -42$. 4. **Write the final answer:** Quotient is $x^2 - 3x + 11$ and remainder is $-42$. 5. **Express the division result:** $$\frac{x^3 + x^2 - x + 2}{x + 4} = x^2 - 3x + 11 + \frac{-42}{x + 4}.$$