1. **State the problem:** Divide the polynomial $$2x^3 - 5x^2 + 3x + 7$$ by the binomial $$x - 2$$. 2. **Formula and method:** Use polynomial long division or synthetic division to divide. Here, we use polynomial long division. 3. **Set up the division:** $$\frac{2x^3 - 5x^2 + 3x + 7}{x - 2}$$ 4. **Divide the leading terms:** Divide $$2x^3$$ by $$x$$ to get $$2x^2$$. 5. **Multiply and subtract:** Multiply $$2x^2$$ by $$x - 2$$: $$2x^2 \times (x - 2) = 2x^3 - 4x^2$$ Subtract this from the original polynomial: $$\left(2x^3 - 5x^2 + 3x + 7\right) - \left(2x^3 - 4x^2\right) = \cancel{2x^3} - 5x^2 + 3x + 7 - \cancel{2x^3} + 4x^2 = -x^2 + 3x + 7$$ 6. **Repeat the process:** Divide $$-x^2$$ by $$x$$ to get $$-x$$. 7. **Multiply and subtract:** Multiply $$-x$$ by $$x - 2$$: $$-x \times (x - 2) = -x^2 + 2x$$ Subtract this from the current remainder: $$\left(-x^2 + 3x + 7\right) - \left(-x^2 + 2x\right) = \cancel{-x^2} + 3x + 7 - \cancel{-x^2} - 2x = x + 7$$ 8. **Continue:** Divide $$x$$ by $$x$$ to get $$1$$. 9. **Multiply and subtract:** Multiply $$1$$ by $$x - 2$$: $$1 \times (x - 2) = x - 2$$ Subtract this from the current remainder: $$\left(x + 7\right) - \left(x - 2\right) = \cancel{x} + 7 - \cancel{x} + 2 = 9$$ 10. **Conclusion:** The quotient is $$2x^2 - x + 1$$ and the remainder is $$9$$. 11. **Final answer:** $$\frac{2x^3 - 5x^2 + 3x + 7}{x - 2} = 2x^2 - x + 1 + \frac{9}{x - 2}$$