$\begin{array}{l}9 x y z \prod(y+z)-2\left[\sum x^{2}(-x+y+z)\right]\left[\sum x(x+y)(x+z)\right]= \\ 9 x y z \sum y z(y+z)+18 x^{2} y^{2} z^{2}-2\left\{\left[\sum x^{2}(y+z)\right]^{2}-\sum\left(x^{3}\right)^{2}\right\}- \\ 6 x y z \sum x^{2}(-x+y+z)= \\ 2\left[\sum x^{6}+3 x^{2} y^{2} z^{2}-\sum y^{2} z^{2}\left(y^{2}+z^{2}\right)\right]+ \\ x y z\left[2 \sum x^{3}-\sum y z(y+z)\right] \geqslant 0\end{array}$