1. **State the problem:** Perform the polynomial long division of $$2x^4 + 3x^3 + 3x^2 - 5x - 3$$ by $$2x^2 - x - 1$$. 2. **Recall the division process:** Divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by this quotient term, subtract from the dividend, and repeat with the remainder. 3. **Divide leading terms:** $$\frac{2x^4}{2x^2} = x^2$$. 4. **Multiply divisor by $$x^2$$:** $$x^2(2x^2 - x - 1) = 2x^4 - x^3 - x^2$$. 5. **Subtract:** $$\begin{aligned} &(2x^4 + 3x^3 + 3x^2) - (2x^4 - x^3 - x^2) \\ &= 2x^4 - 2x^4 + 3x^3 - (-x^3) + 3x^2 - (-x^2) \\ &= 0 + 4x^3 + 4x^2 \end{aligned}$$ 6. **Bring down the next terms:** The new dividend is $$4x^3 + 4x^2 - 5x - 3$$. 7. **Divide leading terms:** $$\frac{4x^3}{2x^2} = 2x$$. 8. **Multiply divisor by $$2x$$:** $$2x(2x^2 - x - 1) = 4x^3 - 2x^2 - 2x$$. 9. **Subtract:** $$\begin{aligned} &(4x^3 + 4x^2 - 5x) - (4x^3 - 2x^2 - 2x) \\ &= 4x^3 - 4x^3 + 4x^2 - (-2x^2) - 5x - (-2x) \\ &= 0 + 6x^2 - 3x \end{aligned}$$ 10. **Bring down the last term:** The new dividend is $$6x^2 - 3x - 3$$. 11. **Divide leading terms:** $$\frac{6x^2}{2x^2} = 3$$. 12. **Multiply divisor by $$3$$:** $$3(2x^2 - x - 1) = 6x^2 - 3x - 3$$. 13. **Subtract:** $$\begin{aligned} &(6x^2 - 3x - 3) - (6x^2 - 3x - 3) = 0 \end{aligned}$$ 14. **Conclusion:** The quotient is $$x^2 + 2x + 3$$ and the remainder is 0. **Final answer:** $$\frac{2x^4 + 3x^3 + 3x^2 - 5x - 3}{2x^2 - x - 1} = x^2 + 2x + 3$$