Giải
$2\left( {{x}^{4}}+{{y}^{4}}+{{z}^{4}} \right)-{{\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}} \right)}^{2}}-2\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}} \right){{\left( x+y+z \right)}^{2}}+{{\left( x+y+z \right)}^{4}}$

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

$=2\left[ \left( {{x}^{4}}+{{y}^{4}}+{{z}^{4}} \right)-{{\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}} \right)}^{2}} \right]+4.{{\left( xy+yz+zx \right)}^{2}}$

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

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

$=4.(2x{{y}^{2}}z+2xy{{z}^{2}}+2{{x}^{2}}yz)$

$=8xyz\left( x+y+z \right)$