1. Problem: Divide the polynomial $6x^5 - x^4 + 4x^3 - 5x^2 - x - 15$ by $2x^2 - x + 3$. 2. Formula and rules: Use polynomial long division: repeatedly divide the leading term of the current dividend by the leading term of the divisor to obtain each quotient term, multiply the divisor by that term, subtract, and bring down the next term. 3. Step 1 — divide leading terms: $$\frac{6x^5}{2x^2} = \frac{\cancel{2}\cdot 3 x^5}{\cancel{2} x^2} = 3x^3$$ 4. Multiply and subtract: $$(2x^2 - x + 3)\cdot 3x^3 = 6x^5 - 3x^4 + 9x^3$$ 5. Subtract from the dividend and bring down the next term: $$(6x^5 - x^4 + 4x^3) - (6x^5 - 3x^4 + 9x^3) = 2x^4 - 5x^3$$ 6. Bring down $-5x^2$ to continue: current dividend becomes $2x^4 - 5x^3 - 5x^2$. 7. Step 2 — divide leading terms: $$\frac{2x^4}{2x^2} = \frac{\cancel{2} x^4}{\cancel{2} x^2} = x^2$$ 8. Multiply and subtract: $$(2x^2 - x + 3)\cdot x^2 = 2x^4 - x^3 + 3x^2$$ 9. Subtract and bring down the next term: $$(2x^4 - 5x^3 - 5x^2) - (2x^4 - x^3 + 3x^2) = -4x^3 - 8x^2$$ 10. Bring down $-x$ to get $-4x^3 - 8x^2 - x$. 11. Step 3 — divide leading terms: $$\frac{-4x^3}{2x^2} = \frac{\cancel{2}(-2) x^3}{\cancel{2} x^2} = -2x$$ 12. Multiply and subtract: $$(2x^2 - x + 3)\cdot (-2x) = -4x^3 + 2x^2 - 6x$$ 13. Subtract and bring down the final term: $$(-4x^3 - 8x^2 - x) - (-4x^3 + 2x^2 - 6x) = -10x^2 + 5x$$ 14. Bring down $-15$ to get $-10x^2 + 5x - 15$. 15. Step 4 — divide leading terms: $$\frac{-10x^2}{2x^2} = \frac{\cancel{2}(-5) x^2}{\cancel{2} x^2} = -5$$ 16. Multiply and subtract: $$(2x^2 - x + 3)\cdot (-5) = -10x^2 + 5x - 15$$ 17. Subtracting gives remainder $0$, so the division is exact. 18. Final answer: the quotient is $3x^3 + x^2 - 2x - 5$ and the remainder is $0$. 19. Verification (optional): $$6x^5 - x^4 + 4x^3 - 5x^2 - x - 15 = (2x^2 - x + 3)(3x^3 + x^2 - 2x - 5).$$