2011/013) factorization using short cuts.

1) (a+b+c)^3 - a^3-b^3-c^3

we know that (a+b+c)^3 - a^3 is divisible by (a+b+c) – a that is b+ c

and b^3+c^3 by (b+c)

hence b+c is a factor

by symmetry (a+b) and (c+a) are also factors

2) so it is (a+b+c)^3 - a^3-b^3-c^3 = m(a+b)(b+c)(c+a)

now as LHS does not contain a^3 or b^3 or c^3 (as they cancel out) so m has to be a constant

putting a =1 b = 1 and c = 1 we get LHS = 24 and RHS = 8m or m = 3

hence (a+b+c)^3 - a^3-b^3-c^3 = 3(a+b)(b+c)(c+a)