Range of $f(x,y)=\frac{4x^2+(y+2)^2}{x^2+y^2+1}$ I am trying to find the range of this function:
$$f(x,y)=\frac{4x^2+(y+2)^2}{x^2+y^2+1}$$ 
So I think that means I have to find minima and maxima. Using partial derivatives gets messy, so I was wondering if I could do some change of variables to make it easier computationally. But no change of coordinates that I can think of have really simplified it much. If I set $2w=y+2$, then I get a problem below. Am I thinking of the right strategy, or is there something better I could do?

It is $$f(x,y)\le 5$$  if
$$4x^2+(y+2)^2\le 5x^2+5y^2+5$$ if
$$0\le x^2+(2y-1)^2$$ and clearly we get $$f(x,y)\geq 0$$
