1. **State the problem:** Divide the polynomial $$2x^4 - x^3 - 8x^2 + 15x - 6$$ by the binomial $$x + 4$$. 2. **Recall the division formula:** Polynomial division is similar to long division with numbers. We divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the divisor by that result, subtract, and repeat. 3. **Set up the division:** Dividend: $$2x^4 - x^3 - 8x^2 + 15x - 6$$ Divisor: $$x + 4$$ 4. **Divide the leading terms:** $$\frac{2x^4}{x} = 2x^3$$ 5. **Multiply divisor by this term:** $$2x^3 \times (x + 4) = 2x^4 + 8x^3$$ 6. **Subtract this from the dividend:** $$\left(2x^4 - x^3\right) - \left(2x^4 + 8x^3\right) = \cancel{2x^4} - x^3 - \cancel{2x^4} - 8x^3 = -9x^3$$ Bring down the next term $$-8x^2$$ to get: $$-9x^3 - 8x^2$$ 7. **Divide leading terms again:** $$\frac{-9x^3}{x} = -9x^2$$ 8. **Multiply divisor by this term:** $$-9x^2 \times (x + 4) = -9x^3 - 36x^2$$ 9. **Subtract:** $$(-9x^3 - 8x^2) - (-9x^3 - 36x^2) = \cancel{-9x^3} - 8x^2 - \cancel{-9x^3} + 36x^2 = 28x^2$$ Bring down the next term $$+15x$$: $$28x^2 + 15x$$ 10. **Divide leading terms:** $$\frac{28x^2}{x} = 28x$$ 11. **Multiply divisor:** $$28x \times (x + 4) = 28x^2 + 112x$$ 12. **Subtract:** $$ (28x^2 + 15x) - (28x^2 + 112x) = \cancel{28x^2} + 15x - \cancel{28x^2} - 112x = -97x$$ Bring down the last term $$-6$$: $$-97x - 6$$ 13. **Divide leading terms:** $$\frac{-97x}{x} = -97$$ 14. **Multiply divisor:** $$-97 \times (x + 4) = -97x - 388$$ 15. **Subtract:** $$(-97x - 6) - (-97x - 388) = \cancel{-97x} - 6 - \cancel{-97x} + 388 = 382$$ 16. **Conclusion:** The quotient is $$2x^3 - 9x^2 + 28x - 97$$ and the remainder is $$382$$. So, $$\frac{2x^4 - x^3 - 8x^2 + 15x - 6}{x + 4} = 2x^3 - 9x^2 + 28x - 97 + \frac{382}{x + 4}$$