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\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=\left(a+b+c\right)\left(ab+bc\right)+\left(a+b+c\right)ac-abc\)
\(=\left(ab+b^2+bc\right)\left(a+c\right)+\left(a+c\right)ac+abc-abc\)
\(=\left(a+c\right)\left(ab+b^2+bc+ac\right)\)
\(=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)
\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=\left(a+b+c\right)\left(ab+bc\right)+\left(a+b+c\right)ac-abc\)
\(=\left(ab+b^2+bc\right)\left(a+c\right)+\left(a+c\right)ac+abc-abc\)
\(=\left(a+c\right)\left(ab+b^2+bc+ac\right)\)
\(=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)
k: \(=a^2b+ab^2-b^2c-bc^2-c^2a+ca^2\)
\(=a^2\left(b+c\right)-bc\left(b+c\right)+a\left(b-c\right)\left(b+c\right)\)
\(=\left(b+c\right)\left(a^2-bc+ab-ac\right)\)
\(=\left(b+c\right)\left[a\left(a+b\right)-c\left(a+b\right)\right]\)
\(=\left(b+c\right)\left(a+b\right)\left(a-c\right)\)
l: \(=a^2b-a^2c+b^2c-b^2a+c^2a-c^2b\)
\(=b\left(a^2-c^2\right)-ac\left(a-c\right)+b^2\left(c-a\right)\)
\(=\left(a-c\right)\left(ba+bc-ac-b^2\right)\)
\(=\left(a-c\right)\left[b\left(a-b\right)+c\left(b-a\right)\right]\)
\(=\left(a-c\right)\left(a-b\right)\left(b-c\right)\)
Ta có : \(A=ab(a-b)+bc(b-c)+ca(c-a)\)
\(\Rightarrow A=ab(a-b)-bc(c-b)+ac(c-a)\)
\(\Rightarrow A=ab(a-b)-bc[(c-a)+(a-b)]+ac(c-a)\)
\(\Rightarrow A=ab(a-b)-bc(a-b)-bc(c-a)+ac(c-a)\)
\(\Rightarrow A=(a-b)(ab-bc)+(c-a)(ac-bc)\)
\(\Rightarrow A=b(a-b)(a-c)-(a-c)c(a-b)\)
\(\Rightarrow A=(a-c)(a-b)(b-c)\)
Chúc học tốt trong kì thi tới :>