(a + b)3 + (b + c)3 + (c + a)3 - 3(a + b)(b + c)(c + a)
= a3 + 3a2b + 3ab2 + b3 + b3 + 3b2c + 3ab2 + c3 + c3 + 3c2a + 3ca2 + a3 - 3ab2 - 3a2b - 6abc - 3ac2 - 3b2c - 3bc2
= 2a3 + 3ac2 - 3ac2 - 6abc + b3 + b3 + c3 + c3 + 3bc2 - 3bc2 + 3b2c - 3b2c
= 2a3 - 6abc + b3 + b3 + c3 + c3 + 3bc2 - 3bc2 + 3b2c - 3b2c
= 2a3 - 6abc + b3 + b3 + c3 + c3
= 2a3 - 6abc + 2b3 + 2c3