(x-1)(x+2)(x-3)(x+4)+24

f(x) = (x-1)(x+2)(x-3)(x+4)+24
f(2) = (1)(4)(-1)(6) + 24 =0
(x-2) is factor of f(x)
f(-3) =(-4)(-1)(-6)(1) + 24 = 0
(x+3) is a factor of f(x)
let
f(x) = (x-2)(x+3)(x^2+ax+b) = (x-1)(x+2)(x-3)(x+4)+24
coef. of constant
-6b= 48
b = -8
put x = 1
-4(1+a-8) = 24
28-4a =24
a= 1
(x-1)(x+2)(x-3)(x+4)+24 = (x-2)(x+3)(x^2+x-8)

(x-1)(x+2)(x-3)(x+4)+24
=(x²+x-2)(x²+x-12)+24
=(x²+x)²-12(x²+x)-2(x²+x)+24+24
=(x²+x)²-14(x²+x)+48
=(x²+x-6)(x²+x-8)
=(x+3)(x-2)(x²+x-8)

(x-1)(x+2)(x-3)(x+4)+24
=(x^2+x-2)(x^2+x-12)+24
=(x^2+x)^2-14(x^2+x)+48
=(x^2+x-6)(x^2+x-8)
=(x-2)(x+3)(x^2+x-8)

(x-1)(x+2)(x-3)(x+4)+24

=(x^2+x-12)(x^2+x-2)+24

=(x^2+x)^2-14(x^2+x)+48

=(x^2+x-6)(x^2+x-8)

=(x+3)(x-2)(x^2+x-8)