1. **Problem:** Divide $3x - 5x^3 + 2$ by $x + 4$ using polynomial long division. 2. **Write the division statement:** $$\frac{-5x^3 + 3x + 2}{x + 4}$$ 3. **Arrange terms in descending powers of $x$:** Dividend: $-5x^3 + 0x^2 + 3x + 2$ Divisor: $x + 4$ 4. **Start division:** Divide the leading term of the dividend $-5x^3$ by the leading term of the divisor $x$: $$\frac{-5x^3}{x} = -5x^2$$ 5. **Multiply divisor by $-5x^2$ and subtract:** $$(-5x^2)(x + 4) = -5x^3 - 20x^2$$ Subtract from dividend: $$(-5x^3 + 0x^2 + 3x + 2) - (-5x^3 - 20x^2) = 0x^3 + 20x^2 + 3x + 2$$ 6. **Next term:** Divide $20x^2$ by $x$: $$\frac{20x^2}{x} = 20x$$ 7. **Multiply divisor by $20x$ and subtract:** $$20x(x + 4) = 20x^2 + 80x$$ Subtract: $$(20x^2 + 3x + 2) - (20x^2 + 80x) = 0x^2 - 77x + 2$$ 8. **Next term:** Divide $-77x$ by $x$: $$\frac{-77x}{x} = -77$$ 9. **Multiply divisor by $-77$ and subtract:** $$-77(x + 4) = -77x - 308$$ Subtract: $$(-77x + 2) - (-77x - 308) = 0x + 310$$ 10. **Remainder:** $310$ 11. **Final answer:** $$-5x^2 + 20x - 77 + \frac{310}{x + 4}$$ --- **Summary:** $$\frac{-5x^3 + 3x + 2}{x + 4} = -5x^2 + 20x - 77 + \frac{310}{x + 4}$$