(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: \[ \boxed{-119m^2 + 52mn - 5n^2}. \]