$$x|x+5|+2|x+7|-2=0$$

$$\begin{aligned} & \text { (i) } d x \geq-5 \Rightarrow x(x+5)+2(x+7)-2=0 \\ & x^2+7 x+12=0 \Rightarrow x=-3,-4 \end{aligned}$$

$$\begin{aligned} \text{(ii)} \quad & x \in(-7,-5) \\ & x(-x-5)+2(x+7)-2=0 \\ & -x^2-3 x+12=0 \\ & \Rightarrow x^2+3 x-12=0 \\ & \Rightarrow x=\frac{-3-\sqrt{57}}{2} \text { satisfy } \end{aligned}$$

$$\begin{aligned} \text{(iii)} \quad & x \leq-7 \\ & \Rightarrow x(-x-5)+2(-x-7)-2=0 \\ & -x^2-7 x-16=0 \Rightarrow x^2+7 x+16=0 \end{aligned}$$

No solution