\(A=4x\left(x+y\right)\left(x+y+z\right)\left(x+z\right)+y^2z^2\)

\(=4\left(x+y\right)\left(x+z\right)x\left(x+y+z\right)+y^2z^2\)

\(=4\left(x^2+xy+xz+yz\right)\left(x^2+xy+xz\right)+y^2z^2\)

Denote \(t=x^2+xy+xz\), then :

\(A=4\left(t+yz\right)t+y^2z^2=4t^2+4yzt+y^2z^2=\left(2t+yz\right)^2\)

\(=\left(2x^2+2xy+2xz+yz\right)^2\ge0\forall x,y,z\)