(a^2+b^2-4c^2)^2-4a^2b^2

Jasonbradley

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Jun 16, 2024

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Factoring the Expression (a^2 + b^2 - 4c^2)^2 - 4a^2b^2

The expression (a^2 + b^2 - 4c^2)^2 - 4a^2b^2 can be factored using the "difference of squares" pattern. Here's how:

Understanding the Difference of Squares Pattern

The difference of squares pattern states that: x^2 - y^2 = (x + y)(x - y)

Applying the Pattern to Our Expression

1. Recognize the Squares:

   • (a^2 + b^2 - 4c^2)^2 is the square of the expression (a^2 + b^2 - 4c^2).
   • 4a^2b^2 is the square of 2ab.

2. Substitute and Factor:

   • Let x = (a^2 + b^2 - 4c^2)
   • Let y = 2ab

   Now we have: x^2 - y^2 = (x + y)(x - y)

   Substitute back: (a^2 + b^2 - 4c^2)^2 - 4a^2b^2 = [(a^2 + b^2 - 4c^2) + 2ab][(a^2 + b^2 - 4c^2) - 2ab]

3. Simplify: (a^2 + b^2 - 4c^2)^2 - 4a^2b^2 = (a^2 + b^2 + 2ab - 4c^2)(a^2 + b^2 - 2ab - 4c^2)

Final Factored Form

Therefore, the factored form of (a^2 + b^2 - 4c^2)^2 - 4a^2b^2 is (a^2 + b^2 + 2ab - 4c^2)(a^2 + b^2 - 2ab - 4c^2).