1. **State the problem:** Expand and simplify the expression $$5m (2m - 5n) - 3n (m + 2n)$$. 2. **Use the distributive property:** Multiply each term inside the brackets by the term outside. 3. **Expand each bracket:** $$5m \times 2m = 10m^2$$ $$5m \times (-5n) = -25mn$$ $$-3n \times m = -3mn$$ $$-3n \times 2n = -6n^2$$ 4. **Write the expanded expression:** $$10m^2 - 25mn - 3mn - 6n^2$$ 5. **Combine like terms:** $$-25mn - 3mn = -28mn$$ 6. **Final simplified expression:** $$10m^2 - 28mn - 6n^2$$