1. **State the problem:** Multiply and simplify the expression $(-2y + 7)(3y - 2)$. 2. **Apply the distributive property (FOIL method):** Multiply each term in the first parenthesis by each term in the second parenthesis. $$(-2y)(3y) + (-2y)(-2) + 7(3y) + 7(-2)$$ 3. **Calculate each product:** $$-6y^2 + 4y + 21y - 14$$ 4. **Combine like terms:** $$-6y^2 + (4y + 21y) - 14 = -6y^2 + 25y - 14$$ 5. **Final simplified expression:** $$\boxed{-6y^2 + 25y - 14}$$