Factorize:
$a^{2} + 6 b x - 9 b^{2} x^{2} - 10 a b - 1 + 25 b^{2}$

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Solution

$a^{2} + 6 b x - 9 b^{2} x^{2} - 10 a b - 1 + 25 b^{2}$
$a^{2} + 6 b x - 9 b^{2} x^{2} - 10 a b + 25 b^{2} - 1$
$a^{2} + 25 b^{2} - 10 a b + 6 b x - 9 b^{2} x^{2} - 1$
$a^{2} + 25 b^{2} - 10 a b - (9 b^{2} x^{2} + 1 - 6 b x)$
${(a - 5 b)}^{2} - {(3 b x - 1)}^{2}$
$(a - 5 b + 3 b x - 1) (a - 5 b - 3 b x + 1)$
Hence $a^{2} + 6 b x - 9 b^{2} x^{2} - 10 a b - 1 + 25 b^{2}$ $=$ $(a - 5 b + 3 b x - 1) (a - 5 b - 3 b x + 1)$

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