Simplify: \(\left[(4x - 5y)^2 + 80xy\right] \div \left[(4x + 5y)^2\right]\)
1
Expand the numerator: \((4x - 5y)^2 = 16x^2 - 40xy + 25y^2\).
Add \(80xy\): \(16x^2 - 40xy + 25y^2 + 80xy = 16x^2 + 40xy + 25y^2\).
This is a perfect square: \(16x^2 + 40xy + 25y^2 = (4x + 5y)^2\).
So the expression becomes \(\dfrac{(4x + 5y)^2}{(4x + 5y)^2} = 1\).
Hence, the simplified value is 1.
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