\({x^4}\left( {y - z} \right) + {y^4}\left( {z - x} \right) + {z^4}\left( {x - y} \right) = \left( {{x^4}y - x{y^4}} \right) + {z^4}\left( {x - y} \right) - z\left( {{x^4} - {y^4}} \right)\)

\( = xy\left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right) + {z^4}\left( {x - y} \right) - z\left( {x - y} \right)\left( {x + y} \right)\left( {{x^2} + {y^2}} \right)\)

\( = \left( {x - y} \right)\left( {{x^3}y + {x^2}{y^2} + x{y^3} + {z^4} - z{x^3} - x{y^2}z - {x^2}yz - {y^3}z} \right)\)

\( = \left( {x - y} \right)\left[ {\left( {{x^3}y - {x^3}z} \right) + \left( {{x^2}{y^2} - {x^2}yz} \right) + \left( {x{y^3} - x{y^2}z} \right) + \left( {{z^4} - {y^3}z} \right)} \right]\)

\( = \left( {x - y} \right)\left[ {{x^3}\left( {y - z} \right) + {x^2}y\left( {y - z} \right) + x{y^2}\left( {y - z} \right) - z\left( {y - z} \right)\left( {{y^2} + yz + {z^2}} \right)} \right]\)

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

\( = \left( {x - y} \right)\left( {y - z} \right)\left[ {\left( {{x^3} - {z^3}} \right) + \left( {{x^2}y - y{z^2}} \right) + \left( {x{y^2} - {y^2}z} \right)} \right]\)

\( = \left( {x - y} \right)\left( {y - z} \right)\left[ {\left( {x - z} \right)\left( {{x^2} + xz + {z^2}} \right) + y\left( {x - z} \right)\left( {x + z} \right) + {y^2}\left( {x - z} \right)} \right]\)

\( = \left( {x - y} \right)\left( {x - z} \right)\left( {y - z} \right)\left( {{x^2} + {y^2} + {z^2} + xy + yz + xz} \right)\).