1. **State the problem:** Expand the expression $$(3x + 2)(2x^2 - 5xy + 4y^2)$$. 2. **Formula used:** Use the distributive property (also called FOIL for binomials) which states that $$a(b + c) = ab + ac$$. Here, distribute each term in the first parenthesis to every term in the second. 3. **Apply distribution:** $$3x \times 2x^2 = 6x^3$$ $$3x \times (-5xy) = -15x^2y$$ $$3x \times 4y^2 = 12xy^2$$ $$2 \times 2x^2 = 4x^2$$ $$2 \times (-5xy) = -10xy$$ $$2 \times 4y^2 = 8y^2$$ 4. **Combine all terms:** $$6x^3 - 15x^2y + 12xy^2 + 4x^2 - 10xy + 8y^2$$ 5. **Group like terms if any:** There are no like terms to combine further. 6. **Final answer:** $$6x^3 - 15x^2y + 4x^2 + 12xy^2 - 10xy + 8y^2$$