Question

factorise:(x+1)(x+2)(x+3)(x+4)+6.

Answer 1

Given Equation:-

(x+1)(x+2)(x+3)(x+4)+6

Solve,

=(x+1)(x+2)(x+3)(x+4)+6
=(x*x+2x+x+2)(x+3)(x+4)+6
=(x²+2x+x+2)(x+3)(x+4)+6
=(x²+3x+2)(x+3)(x+4)+6
=(x³+3x²+3x²+9x+2x+6)(x+4)+6
=(x³+6x²+11x+6)(x+4)+6
=x^4+4x³+6x³+24x²+11x²+44x+6x+24+6
=x^4+10x³+35x²+50x+30
===========================
Answer=>x^4+10x³+35x²+50x+30

Answer 2

(x + 1)(x +2)(x +3)(x +4) + 6

= {(x +1)(x +4) }{(x +2)(x +3) } +6

[put , (x +1)(x +4) = x² + 5x +4
(x +2)(x +3) = x² + 5x +6 ]

= { x² + 5x + 4 }{ x² + 5x + 6 } + 6

Let (x² + 5x) = P

= (P + 4)( P + 6) + 6

= P² + 10P + 24 + 6

= P² + 10P + 30

use, quadratic formula , then,
P = -10 ± √(100 - 30×4)}/2 = ( -5±i√5)

= {P - (-5+ i√5)}{ P -( 5- i√5) }

now , put P = x² + 5x

= {x² + 5x -(-5 + i√5)} {x² + 5x -(-5 -i√5)}