1. The problem is to divide two polynomials using the long division method. 2. The general formula for polynomial long division is: Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient. 3. Multiply the entire divisor by this term and subtract the result from the dividend. 4. Repeat the process with the new polynomial until the degree of the remainder is less than the degree of the divisor. 5. For example, divide $$x^3 + 2x^2 - 5x + 6$$ by $$x - 1$$. 6. Divide the leading term $$x^3$$ by $$x$$ to get $$x^2$$. 7. Multiply $$x^2$$ by $$x - 1$$ to get $$x^3 - x^2$$. 8. Subtract: $$\left(x^3 + 2x^2 - 5x + 6\right) - \left(x^3 - x^2\right) = \cancel{x^3} + 2x^2 - 5x + 6 - \cancel{x^3} + x^2 = 3x^2 - 5x + 6$$. 9. Divide the leading term $$3x^2$$ by $$x$$ to get $$3x$$. 10. Multiply $$3x$$ by $$x - 1$$ to get $$3x^2 - 3x$$. 11. Subtract: $$\left(3x^2 - 5x + 6\right) - \left(3x^2 - 3x\right) = \cancel{3x^2} - 5x + 6 - \cancel{3x^2} + 3x = -2x + 6$$. 12. Divide the leading term $$-2x$$ by $$x$$ to get $$-2$$. 13. Multiply $$-2$$ by $$x - 1$$ to get $$-2x + 2$$. 14. Subtract: $$\left(-2x + 6\right) - \left(-2x + 2\right) = \cancel{-2x} + 6 - \cancel{-2x} - 2 = 4$$. 15. The remainder is $$4$$, which has degree less than the divisor. 16. Therefore, the quotient is $$x^2 + 3x - 2$$ and the remainder is $$4$$. 17. The division result is $$x^2 + 3x - 2 + \frac{4}{x - 1}$$.