1. **State the problem:** Simplify the expression $$\frac{-x^4 + 3x^3 + 5x^2 + x + 4}{-x + 3x + 2}$$. 2. **Simplify the denominator:** Combine like terms in the denominator: $$-x + 3x + 2 = (\cancel{-x} + \cancel{3x}) + 2 = 2x + 2$$. 3. **Factor the denominator:** $$2x + 2 = 2(x + 1)$$. 4. **Rewrite the expression:** $$\frac{-x^4 + 3x^3 + 5x^2 + x + 4}{2(x + 1)}$$. 5. **Factor the numerator if possible:** Try to factor the numerator polynomial. Use polynomial division or synthetic division by $(x+1)$ to check if $(x+1)$ is a factor. 6. **Perform polynomial division:** Divide numerator by $(x+1)$: Set up division: divide $-x^4 + 3x^3 + 5x^2 + x + 4$ by $x+1$. - Leading term division: $-x^4 \div x = -x^3$. - Multiply divisor by $-x^3$: $-x^3(x+1) = -x^4 - x^3$. - Subtract: $( -x^4 + 3x^3 ) - ( -x^4 - x^3 ) = 0 + 4x^3$. - Bring down $5x^2$: new dividend part $4x^3 + 5x^2$. - Divide $4x^3 \div x = 4x^2$. - Multiply divisor by $4x^2$: $4x^2(x+1) = 4x^3 + 4x^2$. - Subtract: $(4x^3 + 5x^2) - (4x^3 + 4x^2) = 0 + x^2$. - Bring down $x$: new dividend part $x^2 + x$. - Divide $x^2 \div x = x$. - Multiply divisor by $x$: $x(x+1) = x^2 + x$. - Subtract: $(x^2 + x) - (x^2 + x) = 0$. - Bring down $+4$. - Divide $4 \div x$ is not possible, so remainder is 4. 7. **Express numerator:** $$-x^4 + 3x^3 + 5x^2 + x + 4 = (x+1)(-x^3 + 4x^2 + x) + 4$$. 8. **Rewrite the original expression:** $$\frac{(x+1)(-x^3 + 4x^2 + x) + 4}{2(x+1)} = \frac{(x+1)(-x^3 + 4x^2 + x)}{2(x+1)} + \frac{4}{2(x+1)}$$. 9. **Cancel common factor $(x+1)$ in the first term:** $$= \frac{\cancel{(x+1)}(-x^3 + 4x^2 + x)}{2\cancel{(x+1)}} + \frac{4}{2(x+1)} = \frac{-x^3 + 4x^2 + x}{2} + \frac{4}{2(x+1)}$$. 10. **Final simplified form:** $$\frac{-x^3 + 4x^2 + x}{2} + \frac{2}{x+1}$$. **Answer:** $$\boxed{\frac{-x^3 + 4x^2 + x}{2} + \frac{2}{x+1}}$$