Factorize the given polynomial expression

$9 {(2 a - b)}^{2} - 4 (2 a - b) - 13$

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Solution

Given that
$9 {(2 a - b)}^{2} - 4 (2 a - b) - 13$

let, $2 a - b = x$
$∴ 9 {(2 a - b)}^{2} - 4 (2 a - b) - 13 = 9 x^{2} - 4 x - 13$

$= 9 x^{2} + 9 x - 13 x - 13$
$= 9 x (x + 1) - 13 (x + 1)$
$= (x + 1) (9 x - 13)$
$= (2 a - b + 1) (9 (2 a - b) - 13)$ $[∵ x = 2 a - b]$
$= (2 a - b + 1) (18 a - 9 b - 13)$

$∴ 9 {(2 a - b)}^{2} - 4 (2 a - b) - 13 = (2 a - b + 1) (18 a - 9 b - 13)$

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