Answer
The required expression is $\underline{\left( x+y \right){{\left( x-y \right)}^{3}}}$.
Work Step by Step
Let us consider the given polynomial:
${{x}^{4}}-{{y}^{4}}-2{{x}^{3}}y+2x{{y}^{3}}$ …………..….. (1)
Factorize it with the help of the technique, ‘‘factoring the polynomial by grouping’’:
$\begin{align}
& {{x}^{4}}-{{y}^{4}}-2{{x}^{3}}y+2x{{y}^{3}}=\left( {{x}^{4}}-{{y}^{4}} \right)+\left( -2{{x}^{3}}y+2x{{y}^{3}} \right) \\
& =\left( {{\left( {{x}^{2}} \right)}^{2}}-{{\left( {{y}^{2}} \right)}^{2}} \right)+\left( -2{{x}^{3}}y+2x{{y}^{3}} \right) \\
& =\left( {{x}^{2}}+{{y}^{2}} \right)\left( {{x}^{2}}-{{y}^{2}} \right)-2xy\left( {{x}^{2}}-{{y}^{2}} \right) \\
& =\left( {{x}^{2}}-{{y}^{2}} \right)\left( {{x}^{2}}+{{y}^{2}}-2xy \right) \\
& =\left( x+y \right)\left( x-y \right){{\left( x-y \right)}^{2}} \\
& =\left( x+y \right){{\left( x-y \right)}^{3}}
\end{align}$
