1. **State the problem:** Divide the polynomial $2x^3 - 9x^2 + x + 14$ by the binomial $x - 2$ using polynomial long division. 2. **Formula and rules:** Polynomial division is similar to numerical long division. We divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by this result, subtract from the dividend, and repeat with the remainder. 3. **Step 1:** Divide the leading term $2x^3$ by $x$ to get $2x^2$. 4. **Step 2:** Multiply $2x^2$ by the divisor $x - 2$: $$2x^2 \times (x - 2) = 2x^3 - 4x^2$$ 5. **Step 3:** Subtract this from the original polynomial: $$\left(2x^3 - 9x^2 + x + 14\right) - \left(2x^3 - 4x^2\right) = \cancel{2x^3} - 9x^2 + x + 14 - \cancel{2x^3} + 4x^2 = -5x^2 + x + 14$$ 6. **Step 4:** Divide the new leading term $-5x^2$ by $x$ to get $-5x$. 7. **Step 5:** Multiply $-5x$ by the divisor: $$-5x \times (x - 2) = -5x^2 + 10x$$ 8. **Step 6:** Subtract this from the current remainder: $$(-5x^2 + x + 14) - (-5x^2 + 10x) = \cancel{-5x^2} + x + 14 - \cancel{-5x^2} - 10x = -9x + 14$$ 9. **Step 7:** Divide the leading term $-9x$ by $x$ to get $-9$. 10. **Step 8:** Multiply $-9$ by the divisor: $$-9 \times (x - 2) = -9x + 18$$ 11. **Step 9:** Subtract this from the current remainder: $$(-9x + 14) - (-9x + 18) = \cancel{-9x} + 14 - \cancel{-9x} - 18 = -4$$ 12. **Step 10:** Since the remainder $-4$ has degree less than the divisor, the division ends here. 13. **Final answer:** The quotient is $$2x^2 - 5x - 9$$ and the remainder is $$-4$$ So, $$\frac{2x^3 - 9x^2 + x + 14}{x - 2} = 2x^2 - 5x - 9 + \frac{-4}{x - 2}$$