1. **State the problem:** Divide the polynomial $$-2x^3 + 5bx^2 - 2b^2x$$ by the binomial $$x - 2b$$. 2. **Recall the formula:** Polynomial division can be done using long division or synthetic division. Here, we use long division. 3. **Set up the division:** $$\frac{-2x^3 + 5bx^2 - 2b^2x}{x - 2b}$$ 4. **Divide the leading term:** Divide $$-2x^3$$ by $$x$$ to get $$-2x^2$$. 5. **Multiply and subtract:** Multiply $$-2x^2$$ by $$x - 2b$$: $$-2x^2 \times (x - 2b) = -2x^3 + 4bx^2$$ Subtract this from the original polynomial: $$(-2x^3 + 5bx^2 - 2b^2x) - (-2x^3 + 4bx^2) = (5bx^2 - 4bx^2) - 2b^2x = bx^2 - 2b^2x$$ 6. **Repeat the process:** Divide the new leading term $$bx^2$$ by $$x$$ to get $$bx$$. 7. **Multiply and subtract:** Multiply $$bx$$ by $$x - 2b$$: $$bx \times (x - 2b) = bx^2 - 2b^2x$$ Subtract this from the current remainder: $$(bx^2 - 2b^2x) - (bx^2 - 2b^2x) = 0$$ 8. **Conclusion:** The remainder is zero, so the quotient is: $$\boxed{-2x^2 + bx}$$ This means: $$\frac{-2x^3 + 5bx^2 - 2b^2x}{x - 2b} = -2x^2 + bx$$