<s>[INST] Range of the function $f(x) = \frac{x^2+14x+9}{x^2+2x+3}\;,$ where $x\in \mathbb{R}$ Calculation of Range of the function $\displaystyle f(x) = \frac{x^2+14x+9}{x^2+2x+3}\;,$ where $x\in \mathbb{R}$
(Can we solve it Using $\bf{A.M\geq G.M}$) Inequality.
$\bf{My\; Try::}$ Let $\displaystyle y = f(x) = \frac{x^2+14x+9}{x^2+2x+3} = \frac{(x+1)^2+12(x+1)-4}{(x+1)^2+1}$
Now Let $(x+1) = t\;,$ where $t\in \mathbb{R}$
So $\displaystyle y=\frac{t^2+12t-9}{t^2+1} = 1+\frac{12t-10}{t^2+1}$
Now I did not understand How can I solve after that
Help me
Thanks
 [/INST] HINT:
Let $\displaystyle y=\frac{x^2+14x+9}{x^2+2x+3}$
Rearrange to form a Quadratic Equation in $x$ 
As $x$ is real, the discriminant must be $\ge0$
</s>