Factorize:
9(2a − b)² − 4(2a − b) − 13

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Solution

We have:
$9 (2 a - b)^{2} - 4 (2 a - b) - 13 Let : (2 a - b) = p$
Thus, the given expression becomes
$9 p^{2} - 4 p - 13$
Now, we must split ( $-$ 4) into two numbers such that their sum is ( $-$ 4) and their product is ( $-$ 117).
Clearly, $- 13 + 9 = - 4 and - 13 \times 9 = - 117$ .
$∴ 9 p^{2} - 4 p - 13 = 9 p^{2} + 9 p - 13 p - 13 = 9 p (p + 1) - 13 (p + 1) = (p + 1) (9 p - 13)$
Putting $p = (2 a - b)$ , we get:
$9 {(2 a - b)}^{2} - 4 (2 a - b) - 13 = [(2 a - b) + 1] [9 (2 a - b) - 13] = (2 a - b + 1) [18 a - 9 b - 13]$

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