x² + 9y² $-$ 25m² $-$ 16n² + 6xy + 40mn

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Solution

$x^{2} + 9 y^{2} - 25 m^{2} - 16 n^{2} + 6 x y + 40 m n = x^{2} + 9 y^{2} + 6 x y - 25 m^{2} - 16 n^{2} + 40 m n = [x^{2} + 6 x y + 9 y^{2}] - [25 m^{2} - 40 m n + 16 n^{2}] = [x^{2} + 2 \times x \times 3 y + {(3 y)}^{2}] - [{(5 m)}^{2} - 2 \times 5 m \times 4 n + {(4 n)}^{2}] = {(x + 3 y)}^{2} - {(5 m - 4 n)}^{2} = [(x + 3 y) + (5 m - 4 n)] [(x + 3 y) - (5 m - 4 n)] = (x + 3 y + 5 m - 4 n) (x + 3 y - 5 m + 4 n)$

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x^{2} + 9 y^{2} - 25 m^{2} - 16 n^{2} + 6 x y + 40 m n

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Factorise Completely : $x^{2} + 6 x y + 9 y^{2} + x + 3 y$

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