\(a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)\)
= \(a^2\left(b-c\right)+b^2c-b^2a+c^2a-c^2b\)
= \(a^2\left(b-c\right)+(b^2c-c^2b)-\left(b^2a-c^2a\right)\)
= \(a^2\left(b-c\right)+bc(b-c)-a\left(b^2-c^2\right)\)
= \(a^2\left(b-c\right)+bc(b-c)-a\left(b^{ }-c\right)\left(b+c\right)\)
= \(a^2\left(b-c\right)+bc(b-c)-\left(b^{ }-c\right)\left(ab+ac\right)\)
= \(\left(b-c\right)\left(a^2+bc-ab-ac\right)\)
= \(\left(b-c\right)\left[\left(a^2-ab\right)-\left(ac-bc\right)\right]\)
= \(\left(b-c\right)\left[a\left(a-b\right)-c\left(a-b\right)\right]\)
= \(\left(b-c\right)\left(a-b\right)\left(a-c\right)\)
\(a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)\)
\(=a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left[\left(a-c\right)-\left(b-c\right)\right]\)\(=a^2\left(b-c\right)+b^2\left(c-a\right)-c^2\left(c-a\right)-c^2\left(b-c\right)\)\(=\left(b-c\right)\left(a-c\right)\left(a+c\right)+\left(c-a\right)\left(b+c\right)\left(b-c\right)=\left(b-c\right)\left(a-c\right)\left(a+c\right)-\left(a-c\right)\left(b+c\right)\left(b-c\right)=\left(b-c\right)\left(a-c\right)\left[\left(a+c\right)-\left(b+c\right)\right]=\left(b-c\right)\left(a-c\right)\left(a-b\right)\)
