$f(x)=\log _4\left(\log _3\left(\log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)\right)\right.$

$$\begin{aligned} & \log _3\left(\log _1\left(8-\log _2\left(x^2+4 x+5\right)\right)\right)>0 \\ & \log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)>1 \\ & 8-\log _2\left(x^2+4 x+5\right)>7 \\ & -\log _2\left(x^2+4 x+5\right)>-1 \\ & \log _2\left(x^2+4 x+5\right)<1 \\ & x^2+4 x+5<2 \\ & x^2+4 x+3<0 \\ & \Rightarrow(x+3)(x+1)<0 \quad \ldots(1) \\ & \log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)>0 \\ & 8-\log _2\left(x^2+4 x+5\right)>1 \\ & \log _2\left(x^2+4 x+5\right)<9 \\ & x^2+4 x+5<2^9 \\ & x^2+4 x+5<512 \\ & \Rightarrow x^2+4 x-507<0 \\ & \Rightarrow x=-4 \pm \sqrt{16+2028} \end{aligned}$$

$$\begin{aligned} & x=\frac{-4 \pm \sqrt{2044}}{2} \quad\text{..... (2)}\\ & \Rightarrow\left(x-\left(\frac{-4+\sqrt{2044}}{2}\right)\right)\left(x-\left(\frac{-4-\sqrt{2044}}{2}\right)\right)<0 \\ & x^2+4 x+5>0 \\ & D>0 \\ & x \in R \\ & \text { Also, } 8-\log _2\left(x^2+4 x+5\right)>0 \\ & \log _2\left(x^2+4 x+5\right)<8 \\ & x^2+4 x+5<256 \\ & \Rightarrow x^2+4 x-251<0 \\ & \Rightarrow x=-4 \pm \sqrt{16+1004} \\ & \Rightarrow x=\frac{-4 \pm \sqrt{1020}}{2} \end{aligned}$$

$$\begin{aligned} &\Rightarrow\left(x-\left(\frac{-4+\sqrt{1020}}{2}\right)\right)\left(x-\left(\frac{-4-\sqrt{1020}}{2}\right)\right)<0\\ &\therefore \text { Intersection of (1), (2) and (3) } \end{aligned}$$

$$\begin{aligned} & \therefore x \in(-3,-1) \\ & -1 \leq \frac{7 x+10}{x-2} \leq 1 \\ & \Rightarrow x \in[-2,-1] \\ & \therefore \alpha^2+\beta^2+\gamma^2+\delta^2=(-3)^2+(-1)^2+(-2)^{-2}+(-1)^2 \\ & =9+1+4+1 \\ & =15 \end{aligned}$$