Factorise:

$(x^{2} + 4 y^{2} - 9 z^{2})^{2} - 16 x^{2} y^{2}$

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Solution

$(x^{2} + 4 y^{2} - 9 z^{2})^{2} - 16 x^{2} y^{2} = (x^{2} + (2 y)^{2} - (3 z)^{2})^{2} - (4 x y)^{2}, using identity (a^{2} - b^{2}) = (a + b) (a - b) = (x^{2} + (2 y)^{2} - (3 z)^{2} - 4 x y) (x^{2} + (2 y)^{2} - (3 z)^{2} + 4 x y) = (x^{2} + (2 y)^{2} - 4 x y - (3 z)^{2}) (x^{2} + (2 y)^{2} + 4 x y - (3 z)^{2}) = ((x - 2 y)^{2} - (3 z)^{2}) ((x + 2 y)^{2} - (3 z)^{2}) = [(x - 2 y - 3 z) (x - 2 y + 3 z)] [(x + 2 y - 3 z) (x + 2 y + 3 z)] = (x - 2 y - 3 z) (x - 2 y + 3 z) (x + 2 y - 3 z) (x + 2 y + 3 z)$

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