Sophie Germain's Identity/Proof 1

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Theorem

$x^4 + 4 y^4 = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}$


Proof

\(\ds \) \(\) \(\ds \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}\)
\(\ds \) \(=\) \(\ds x^4 + x^2 \cdot 2 y^2 - x^2 \cdot 2 x y + x^2 \cdot 2 y^2 + 4 y^2 - 2 y^2 \cdot 2 x y + x^2 \cdot 2 x y + 2 y^2 \cdot 2 x y - 2 x y \cdot 2 x y\)
\(\ds \) \(=\) \(\ds x^4 + 4 y^4\) gathering up terms and cancelling

$\blacksquare$