factorise:
1) x^4+10x^3+35x^2+50x+24
2) 3(a-b)(b-c)(c-a)
3) (a+b)^3-a-b
1. use the factor theorem
looking at 24, I tried ±1, ±2, ...
and found that x=-1 and x=-2 worked
so I used synthetic division consecutively and found
x^4+10x^3+35x^2+50x+24
= (x+1)(x+2)(x^2 + 7x + 12)
and the quadratic factors again, so the final factors are
(x+1)(x+2)(x+3)(x+4)
2. already fully factored
3. (a+b)^3 - a - b
= (a+b)^3 - (a+b)
= (a+b)( (a+b)^2 - 1) , last part is difference of squares
= (a+b)(a+b+1)(a+b-1)