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 quotient, subtract, 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: $$\left(2x^3 + 9x^2 + x + 14\right) - \left(2x^3 - 4x^2\right) = \cancel{2x^3} + 9x^2 + x + 14 - \cancel{2x^3} + 4x^2 = 13x^2 + x + 14$$ 6. **Step 4:** Divide the new leading term $13x^2$ by $x$ to get $13x$. 7. **Step 5:** Multiply $13x$ by the divisor: $$13x \times (x - 2) = 13x^2 - 26x$$ 8. **Step 6:** Subtract this from the current remainder: $$\left(13x^2 + x + 14\right) - \left(13x^2 - 26x\right) = \cancel{13x^2} + x + 14 - \cancel{13x^2} + 26x = 27x + 14$$ 9. **Step 7:** Divide $27x$ by $x$ to get $27$. 10. **Step 8:** Multiply $27$ by the divisor: $$27 \times (x - 2) = 27x - 54$$ 11. **Step 9:** Subtract this from the current remainder: $$\left(27x + 14\right) - \left(27x - 54\right) = \cancel{27x} + 14 - \cancel{27x} + 54 = 68$$ 12. **Step 10:** Since the remainder $68$ is a constant and the divisor is degree 1, division stops here. 13. **Final answer:** $$\frac{2x^3 + 9x^2 + x + 14}{x - 2} = 2x^2 + 13x + 27 + \frac{68}{x - 2}$$