1. **State the problem:** Divide the polynomial $$x^4 - 3x^3 - 7x - 14$$ by $$x - 4$$. 2. **Formula and rule:** 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, then multiply and subtract repeatedly. 3. **Set up the division:** Dividend: $$x^4 - 3x^3 + 0x^2 - 7x - 14$$ (note the zero coefficient for $$x^2$$) Divisor: $$x - 4$$ 4. **Divide the leading terms:** $$\frac{x^4}{x} = x^3$$ 5. **Multiply divisor by $$x^3$$:** $$x^3 \times (x - 4) = x^4 - 4x^3$$ 6. **Subtract:** $$\left(x^4 - 3x^3 + 0x^2 - 7x - 14\right) - \left(x^4 - 4x^3\right) = (x^4 - x^4) + (-3x^3 + 4x^3) + 0x^2 - 7x - 14 = 0 + x^3 + 0x^2 - 7x - 14$$ 7. **Bring down the next term:** Now divide $$x^3$$ by $$x$$: $$\frac{x^3}{x} = x^2$$ 8. **Multiply divisor by $$x^2$$:** $$x^2 \times (x - 4) = x^3 - 4x^2$$ 9. **Subtract:** $$\left(x^3 + 0x^2 - 7x - 14\right) - \left(x^3 - 4x^2\right) = (x^3 - x^3) + (0x^2 + 4x^2) - 7x - 14 = 0 + 4x^2 - 7x - 14$$ 10. **Divide $$4x^2$$ by $$x$$:** $$\frac{4x^2}{x} = 4x$$ 11. **Multiply divisor by $$4x$$:** $$4x \times (x - 4) = 4x^2 - 16x$$ 12. **Subtract:** $$\left(4x^2 - 7x - 14\right) - \left(4x^2 - 16x\right) = (4x^2 - 4x^2) + (-7x + 16x) - 14 = 0 + 9x - 14$$ 13. **Divide $$9x$$ by $$x$$:** $$\frac{9x}{x} = 9$$ 14. **Multiply divisor by 9:** $$9 \times (x - 4) = 9x - 36$$ 15. **Subtract:** $$\left(9x - 14\right) - \left(9x - 36\right) = (9x - 9x) + (-14 + 36) = 0 + 22$$ 16. **Remainder:** The remainder is 22, which is less than the degree of the divisor. 17. **Final answer:** $$\frac{x^4 - 3x^3 - 7x - 14}{x - 4} = x^3 + x^2 + 4x + 9 + \frac{22}{x - 4}$$ **Note:** The quotient given in the problem statement $$x^3 + 1x^2 + 4x + 1$$ with remainder 10 is incorrect based on the division steps above. **Summary:** The correct quotient is $$x^3 + x^2 + 4x + 9$$ with remainder 22.