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a)a(b2+c2)+b(a2+c2)+c(a2+b2)+2abc
=ab2+ac2+ba2+bc2+ca2+cb2+2abc
=(ab2+ba2)+(ac2+bc2)+(ca2+abc)+(cb2+abc)
=ab(a+b)+c2(a+b)+ca(a+b)+cb(a+b)
=(a+b)(ab+c2+ca+cb)
=(a+b)(a+c)(b+c)
b)a3-b3-c3-3abc
=(a-b)3-c3+3ab(a-b)-3abc
=(a-b-c)[(a-b)2+(a-b)c+c2]+3ab(a-b-c)
=(a-b-c)(a2-2ab+b2+ac-bc+c2+3ab)
=(a-b-c)(a2+b2+c2+ab-bc+ca)
c) \(a^2b^2\left(a-b\right)+b^2c^2\left(b-c\right)+c^2a^2\left(c-a\right)\)
\(=a^2b^2\left(a-b\right)+b^2c^2\left(b-c\right)-c^2a^2\left[\left(a-b\right)+\left(b-c\right)\right]\)
\(=a^2b^2\left(a-b\right)+b^2c^2\left(b-c\right)-c^2a^2\left(a-b\right)-c^2a^2\left(b-c\right)\)
\(=\left(a-b\right)\left(a^2b^2-c^2a^2\right)+\left(b-c\right)\left(b^2c^2-c^2a^2\right)\)
\(=a^2\left(a-b\right)\left(b-c\right)\left(b+c\right)+c^2\left(b-c\right)\left(b-a\right)\left(a+b\right)\)
\(=\left(a-b\right)\left(b-c\right)\left[a^2\left(b+c\right)-c^2\left(a+b\right)\right]\)
\(=\left(a-b\right)\left(b-c\right)\left(a-c\right)\left(ab+bc+ca\right)\)