Finding range of a polynomial expression.

Question

Consider $ y=\frac{2x}{1+x^2}$ , where x is real , then find the range of the expression $y^2 + y - 2$.

I solved it by factorising the expression and then simplifying $$y^2 + y - 2$$ $$= (y+2)(y-1)$$ $$= (\frac{2x}{1+x^2}+2)(\frac{2x}{1+x^2}-1)$$ $$= -2(\frac{(1+x)^2(1+x+x^2)}{(1+x^2)^2}$$

I cannot solve further from here as I can't find range of this expression.

Answer 1

$$y^2+y-2=\dfrac{(2y+1)^2-9}4$$

WLOG $x=\tan A\implies y=\sin2A$

$2y+1=2\sin2A+1$

and $-2+1\le2\sin2A+1\le2+1$

Answer 2

Hint:

Find the range of $y$ first. Say it is $[\alpha,\beta\kern 1.5mu]$. Then find the image of the interval $[\alpha,\beta\mkern 1.5mu]$ by the function $y^2+y-2$.