(5m-2n)²-9(4m-n)²

To simplify the expression \((5m - 2n)^2 - 9(4m - n)^2\), we can start by expanding each square. 1. **Expand \((5m - 2n)^2\)**: \[ (5m - 2n)^2 = (5m)^2 - 2 \cdot (5m)(2n) + (2n)^2 = 25m^2 - 20mn + 4n^2. \] 2. **Expand \(9(4m - n)^2\)**: First, calculate \((4m - n)^2\): \[ (4m - n)^2 = (4m)^2 - 2 \cdot (4m)(n) + (n)^2 = 16m^2 - 8mn + n^2. \] Now, multiply by 9: \[ 9(4m - n)^2 = 9(16m^2 - 8mn + n^2) = 144m^2 - 72mn + 9n^2. \] 3. **Now, substitute back into the original expression**: \[ (5m - 2n)^2 - 9(4m - n)^2 = (25m^2 - 20mn + 4n^2) - (144m^2 - 72mn + 9n^2). \] 4. **Distribute the negative sign**: \[ = 25m^2 - 20mn + 4n^2 - 144m^2 + 72mn - 9n^2. \] 5. **Combine like terms**: - For \(m^2\) terms: \(25m^2 - 144m^2 = -119m^2\). - For \(mn\) terms: \(-20mn + 72mn = 52mn\). - For \(n^2\) terms: \(4n^2 - 9n^2 = -5n^2\). Putting it all together, we get: \[ -119m^2 + 52mn - 5n^2. \] Thus, the simplified expression is: \[ \boxed{-119m^2 + 52mn - 5n^2}. \]