1. **State the problem:** We need to divide the polynomial $$3x^3 + 5x^2 + 10x - 4$$ by the binomial $$3x - 1$$. 2. **Formula and rules:** Polynomial division can be done using long division or synthetic division. Here, we use long division. 3. **Set up the division:** Divide the leading term of the numerator by the leading term of the denominator: $$\frac{3x^3}{3x} = x^2$$ 4. **Multiply and subtract:** Multiply $$x^2$$ by the divisor $$3x - 1$$: $$x^2(3x - 1) = 3x^3 - x^2$$ Subtract this from the original polynomial: $$\left(3x^3 + 5x^2 + 10x - 4\right) - \left(3x^3 - x^2\right) = 6x^2 + 10x - 4$$ 5. **Repeat the process:** Divide the new leading term by the leading term of the divisor: $$\frac{6x^2}{3x} = 2x$$ 6. **Multiply and subtract:** $$2x(3x - 1) = 6x^2 - 2x$$ Subtract: $$\left(6x^2 + 10x - 4\right) - \left(6x^2 - 2x\right) = 12x - 4$$ 7. **Repeat again:** Divide the leading term: $$\frac{12x}{3x} = 4$$ 8. **Multiply and subtract:** $$4(3x - 1) = 12x - 4$$ Subtract: $$\left(12x - 4\right) - \left(12x - 4\right) = 0$$ 9. **Conclusion:** The quotient is: $$x^2 + 2x + 4$$ No remainder remains. **Final answer:** $$\boxed{x^2 + 2x + 4}$$