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Prove that `(a+b+c)^(3)-a^(3)-b^(3)-c^(3)=3(a+b)(b+c)(c+a)`.

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


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