1. **State the problem:** Divide the polynomial $$-8x^3 - 6x^2 + 15x + 11$$ by the binomial $$2x + 3$$ using long division. 2. **Recall the formula and rules:** Polynomial long 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 term, subtract from the dividend, and repeat with the remainder. 3. **Step 1:** Divide the leading term $$-8x^3$$ by $$2x$$: $$\frac{-8x^3}{2x} = -4x^2$$ 4. **Step 2:** Multiply the divisor by $$-4x^2$$: $$-4x^2 \times (2x + 3) = -8x^3 - 12x^2$$ 5. **Step 3:** Subtract this from the original polynomial: $$(-8x^3 - 6x^2 + 15x + 11) - (-8x^3 - 12x^2) = (-8x^3 + 8x^3) + (-6x^2 + 12x^2) + 15x + 11 = 6x^2 + 15x + 11$$ 6. **Step 4:** Divide the new leading term $$6x^2$$ by $$2x$$: $$\frac{6x^2}{2x} = 3x$$ 7. **Step 5:** Multiply the divisor by $$3x$$: $$3x \times (2x + 3) = 6x^2 + 9x$$ 8. **Step 6:** Subtract this from the current remainder: $$ (6x^2 + 15x + 11) - (6x^2 + 9x) = (6x^2 - 6x^2) + (15x - 9x) + 11 = 6x + 11$$ 9. **Step 7:** Divide the new leading term $$6x$$ by $$2x$$: $$\frac{6x}{2x} = 3$$ 10. **Step 8:** Multiply the divisor by $$3$$: $$3 \times (2x + 3) = 6x + 9$$ 11. **Step 9:** Subtract this from the current remainder: $$ (6x + 11) - (6x + 9) = (6x - 6x) + (11 - 9) = 2$$ 12. **Step 10:** Since the remainder $$2$$ is of lower degree than the divisor, the division ends here. 13. **Write the quotient and remainder:** $$\text{Quotient} = -4x^2 + 3x + 3$$ $$\text{Remainder} = 2$$ 14. **Express the final answer:** $$\frac{-8x^3 - 6x^2 + 15x + 11}{2x + 3} = -4x^2 + 3x + 3 + \frac{2}{2x + 3}$$ 15. **Match with given options:** This corresponds to option C. **Final answer:** $$\boxed{-4x^2 + 3x + 3 + \frac{2}{2x + 3}}$$