$(a + b + c)^3 - a^3 - b^3 - c^3$
$= (a + b +c - a)[(a+b+c)^2 + a(a+b+c) + a^2] - (b+c)(b^2 - bc + c^2)$
$= (b+c)(a^2 + b^2 + c^2 + 2ab + 2bc + 2ca + a^2+ ab+ ac + a^2 - b^2 + bc - c^2)$
$= (b+c)(3a^2 + 3ab + 3bc + 3ac))$
$= 3(b+c)[a(a + b) + c(a + b)]$
$= 3(a+b)(b+c)(c+a)$