Question:
Factorize: \[ x(x^3-y^3)+3xy(x-y) \]
Solution:
\[ x(x^3-y^3)+3xy(x-y) \]
\[ =x(x-y)(x^2+xy+y^2)+3xy(x-y) \]
\[ =(x-y)\left[x(x^2+xy+y^2)+3xy\right] \]
\[ =(x-y)(x^3+x^2y+xy^2+3xy) \]
\[ =(x-y)\left(x^3+x^2y+xy^2+y^3-y^3+3xy\right) \]
\[ =(x-y)\left[(x+y)(x^2+y^2)+xy(x+y)-y^3+3xy\right] \]
\[ =(x-y)(x+y)(x^2+y^2+xy) \]
\[ \boxed{(x-y)(x+y)(x^2+xy+y^2)} \]