\(x^4y^4+16+\left(xy+2\right)^4=t^4+16+\left(t+2\right)^4\)(\(t=xy\))
\(=t^4+16+\left(t^2+4t+4\right)^2=t^4+16+\left(t^2+2t+4\right)^2+4t\left(t^2+2t+4\right)+4t^2\)
\(=\left(t^2+2t+4\right)^2+t^4+4t^3+12t^2+16t+16\)
\(=2\left(t^2+2t+4\right)^2\)
\(=2\left(x^2y^2+2xy+4\right)^2\)