Factorize:

We have:

$x^{3}-2 x^{2} y+3 x y^{2}-6 y^{3}=\left(x^{3}-2 x^{2} y\right)+\left(3 x y^{2}-6 y^{3}\right)$

$=x^{2}(x-2 y)+3 y^{2}(x-2 y)$

$=(x-2 y)\left(x^{2}+3 y^{2}\right)$