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a^{2}\left(a-b\right)+b^{2}\left(a-b\right)
Do the grouping a^{3}-a^{2}b+ab^{2}-b^{3}=\left(a^{3}-a^{2}b\right)+\left(ab^{2}-b^{3}\right), and factor out a^{2} in the first and b^{2} in the second group.
\left(a-b\right)\left(a^{2}+b^{2}\right)
Factor out common term a-b by using distributive property.