$(a-1)\left(x^2+\sqrt{3} x+1\right)^2-(a+1)\left[\left(x^2+1\right)^2-(x \sqrt{3})^2\right) \leq 0 $
$ \text { or }(a-1)\left(x^2+\sqrt{3} x+1\right)^2-(a+1)\left[x^2+x \sqrt{3}+1\right) $
$\left(x^2-x \sqrt{3}+1\right) \leq 0 $
$ \left(x^2-x \sqrt{3}+1\right)\left[(a-1)\left(x^2+\sqrt{3} x+1\right)-(a+1)\right. $ $\left.\left(x^2-x \sqrt{3}+1\right)\right] \leq 0 \forall x \in R $
$ \Rightarrow-2\left(x^2+1\right)+2 a \sqrt{3} x \leq 0 $
$ \Rightarrow x^2-a \sqrt{3} x+1 \geq 0 \forall x \in R $
$ \Rightarrow 3 a^2-4 \leq 0(D \leq 0) $
$ \Rightarrow a \in\left[-\frac{2}{\sqrt{3}}, \frac{2}{\sqrt{3}}\right]$
$\Rightarrow \text { Number of possible integral value of a is }\{-1,0,1\} $
$\Rightarrow 3$
and sum of all integral values of a is $-1+0+1=0$
