1. **State the problem:** Simplify the expression $4d(d - 5f) + 2f(3d + 7f)$. 2. **Apply the distributive property:** Multiply each term inside the parentheses by the factor outside. $$4d(d - 5f) = 4d \times d - 4d \times 5f = 4d^2 - 20df$$ $$2f(3d + 7f) = 2f \times 3d + 2f \times 7f = 6df + 14f^2$$ 3. **Combine the results:** $$4d^2 - 20df + 6df + 14f^2$$ 4. **Combine like terms:** The terms $-20df$ and $6df$ are like terms. $$-20df + 6df = \cancel{-20df} + \cancel{6df} = -14df$$ 5. **Write the simplified expression:** $$4d^2 - 14df + 14f^2$$ **Final answer:** $4d^2 - 14df + 14f^2$