Pinko wrote: If $x$ and $y$ are real numbers, determine the greatest possible value of the expression $\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)}$. We have: $$(x^2+1)^2(y^2+1)^2=(x^2+1)(y^2+1)(x^2+1)(y^2+1) \geq \frac{(x+1)^2(y+1)^2(xy+1)^2}{4}$$Therefore: $$\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)} \leq 2$$Equality at $x=y=1$

If $x$ and $y$ are real numbers, determine the greatest possible value of the expression $\frac{(x+1)(y+1)(3xy+1)}{(x^2+1)(y^2+1)}.$