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?

Idea: $$f(x,y) =\frac{4x^2+4y^2+4+(y+2)^2-4y^2-4}{x^2+y^2+1}$$
$$=4+\frac{-3y^2+4y}{x^2+y^2+1}$$
$$\leq 4+\frac{-3y^2+4y}{y^2+1}$$ if $-3y^2+4y\geq 0$ (else it is reversed),
$$= 1+\frac{4y+3}{y^2+1} =:g(y)$$
So you have to find a maximum value of $g$ on $[0,{4\over 3}]$.
