Question

Please factorise-
(x+1)(x+2)(x+3)(x+4)-15
Try and answer quickly plz.
The answer is (x²+5x+1)(x²+5x+9)
But I can't find out how.
Thanks!

Answer 1

Hi...☺

Here is your answer...✌

(x+1) (x+2) (x+3) (x+4) - 15

= [ (x+1) (x+4) ] [ (x+2) (x+3) ] - 15

= (x² + 5x + 4) (x² + 5x + 6) - 15

= [ (x²+5x+5) - 1 ] [( x²+5x+5) + 1 ] - 15

= (x² + 5x + 5)² - 1² - 15
$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [ \because(a - b)(a + b) = {a}^{2} - {b}^{2} ]$

= (x² + 5x + 5)² - 1 - 15

= (x² + 5x + 5)² - 16

= (x² + 5x + 5)² - 4²

= (x² + 5x + 5 + 4) (x² + 5x + 5 - 4)

= ( x² + 5x + 9 ) ( x² + 5x + 1 )