Factorise ${(x^{2} - 2 x)}^{2} - 23 (x^{2} - 2 x) + 120$ .

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Solution

We introduce $a = x^{2} - 2 x$ . Then the expression is $a^{2} - 23 a + 120$ . We observe that $- 23 = (- 15) + (- 8)$ and $120 = (- 15) (- 8)$ . Hence we may write
$a^{2} - 23 a + 120 = a^{2} - 15 a - 8 a + 120 = a (a - 15) - 8 (a - 15) = (a - 15) (a - 8)$
Hence
${(x^{2} - 2 x)}^{2} - 23 (x^{2} - 2 x) + 120 = (x^{2} - 2 x - 15) (x^{2} - 2 x - 8)$
Further we see
$x^{2} - 2 x - 15 = x^{2} - 5 x + 3 x - 15 = x (x - 5) + 3 (x - 5) = (x + 3) (x - 5)$
$x^{2} - 2 x - 8 = x (x - 4) + 2 (x - 4) = (x + 2) (x - 4)$
We obtain
${(x^{2} - 2 x)}^{2} - 23 (x^{2} - 2 x) + 120 = (x + 3) (x - 5) (x + 2) (x - 4)$

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