If x is real and $y=\frac{x^2+2x+p}{x^2+4x+3}$ takes all real values such that $a<p<b$ and product of A.M and G.M of $\left|a\right|,\left|b\right|$ is $k\sqrt{3}$ then k =

$y=\frac{x^2+2x+p}{x^2+4x+3},x\in R$

$(y-1)x^2+2(2y-1)x+(3y-p)=0$

$\Delta \ge 0\Rightarrow 4{(2y-1)}^2-4(y-1)(3y-p)\ge 0$

$x\in R\Rightarrow \Delta \le 0\Rightarrow (p-1)(p+3)\le 0$

$\therefore -3<p<1$

$\left|a\right|=3\left|b\right|=1$

$\left(Am\right)\left(Gm\right)=\left(\frac{3+1}{2}\right)\sqrt{3.1}=2\sqrt{3}$