Factorize $25 (a + 2 b - 3 c)^{2} - 9 (2 a - b - c)^{2}$

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Solution

$25 (a + 2 b - 3 c)^{2} - 9 (2 a - b - c)^{2} = [5 (a + 2 b - 3 c)]^{2} - [3 (2 a - b - c)]^{2}$
$= [5 (a + 2 b - 3 c) + 3 (2 a - b - c)] [5 (a + 2 b - 3 c) - 3 (2 a - b - c)]$
$= (5 a + 10 b - 15 c + 6 a - 3 b - 3 c) (5 a + 10 b - 15 c - 6 a + 3 b + 3 c)$
$= (11 a + 7 b - 18 c) (- a + 13 b - 12 c)$

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