1. **State the problem:** Find the product and simplify the expression $$(4w + 3)(-3w^2 - 3w - 3).$$ 2. **Recall the distributive property:** To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial and then combine like terms. 3. **Multiply each term:** $$4w \times (-3w^2) = -12w^3$$ $$4w \times (-3w) = -12w^2$$ $$4w \times (-3) = -12w$$ $$3 \times (-3w^2) = -9w^2$$ $$3 \times (-3w) = -9w$$ $$3 \times (-3) = -9$$ 4. **Write the expanded expression:** $$-12w^3 - 12w^2 - 12w - 9w^2 - 9w - 9$$ 5. **Combine like terms:** $$-12w^3 + (-12w^2 - 9w^2) + (-12w - 9w) - 9$$ $$= -12w^3 - 21w^2 - 21w - 9$$ 6. **Final simplified answer:** $$\boxed{-12w^3 - 21w^2 - 21w - 9}$$