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\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=\left(a+b+c\right)\left(ab+bc\right)+\left(a+b+c\right)ac-abc\)
\(=\left(ab+b^2+bc\right)\left(a+c\right)+\left(a+c\right)ac+abc-abc\)
\(=\left(a+c\right)\left(ab+b^2+bc+ac\right)\)
\(=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)
b: \(=\left(a+b\right)\left(ab+bc+ca\right)+c\left(ab+bc+ca\right)-abc\)
\(=\left(a+b\right)\left(ab+bc+ca\right)+abc+c\left(bc+ca\right)-abc\)
\(=\left(a+b\right)\cdot\left(ab+ac+bc\right)+c^2\left(a+b\right)\)
\(=\left(a+b\right)\left(ab+ac+bc+c^2\right)\)
\(=\left(a+b\right)\left[a\left(b+c\right)+c\left(b+c\right)\right]\)
\(=\left(a+b\right)\cdot\left(b+c\right)\left(a+c\right)\)
c:\(=a\left(a^3+6a^2b+12ab^2+8b^3\right)-b\left(8a^3+12a^2b+6ab^2+b^3\right)\)
\(=a^4+6a^3b+12a^2b^2+8ab^3-8a^3b-12a^2b^2-6ab^3-b^4\)
\(=a^4-b^4+6a^3b-6ab^3+8ab^3-8a^3b\)
\(=\left(a-b\right)\left(a+b\right)\left(a^2+b^2\right)+6ab\left(a-b\right)\left(a+b\right)+8ab\left(b-a\right)\left(b+a\right)\)
\(=\left(a-b\right)\left(a+b\right)\left(a^2+b^2+6ab-8ab\right)\)
\(=\left(a-b\right)^3\cdot\left(a+b\right)\)
\(a\left(b^2+c^2+bc\right)+b\left(c^2+a^2+ac\right)+c\left(a^2+b^2+ab\right)\)
\(=ab^2+ac^2+abc+bc^2+ba^2+abc+ac^2+bc^2+abc\)
\(=c^2\left(b+a\right)+\left(b^2+3\text{a}b+a^2\right)c+ab^2+a^2b\)
\(=bc^2+ac^2+b^2c+3\text{a}bc+a^2c+ab^2+a^2b\)
\(=\left(c+b+a\right)\left(bc+ac+ab\right)\)
\(\left(a+b+c\right)\left(ab+bc+ca\right)-abc\)
\(=\left(a+b+c\right)\left(ab+bc\right)+\left(a+b+c\right)ac-abc\)
\(=\left(ab+b^2+bc\right)\left(a+c\right)+\left(a+c\right)ac+abc-abc\)
\(=\left(a+c\right)\left(ab+b^2+bc+ac\right)\)
\(=\left(a+b\right)\left(b+c\right)\left(c+a\right)\)