\(C=\frac{\left(x+y+2\right)^2}{xy+2\left(x+y\right)}+\frac{xy+2\left(x+y\right)}{\left(x+y+2\right)^2}=\frac{8}{9}.\frac{\left(x+y+2\right)^2}{xy+2\left(x+y\right)}+\frac{\left(x+y+2\right)^2}{9\left(xy+2x+2y\right)}+\frac{xy+2x+2y}{\left(x+y+2\right)^2}\)
\(C\ge\frac{4}{9}.\frac{2x^2+2y^2+4xy+8x+8x+8}{xy+2x+2y}+2\sqrt{\frac{\left(x+y+2\right)^2\left(xy+2x+2y\right)}{9\left(xy+2x+2y\right)\left(x+y+2\right)^2}}\)
\(C\ge\frac{4}{9}.\frac{\left(x^2+y^2\right)+\left(x^2+4\right)+\left(y^2+4\right)+4xy+8x+8y}{xy+2x+2y}+\frac{2}{3}\)
\(C\ge\frac{4}{9}.\frac{2xy+4x+4y+4xy+8x+8y}{xy+2x+2y}+\frac{2}{3}\)
\(C\ge\frac{4}{9}.\frac{6\left(xy+2x+2y\right)}{xy+2x+2y}+\frac{2}{3}=\frac{8}{3}+\frac{2}{3}=\frac{10}{3}\)
\(C_{min}=\frac{10}{3}\) khi \(x=y=2\)
