$$|x| \quad|x+2|-5|x+1|-1=0$$

$$\begin{aligned} & \text { (I) if } x<-2 \\ & x^2+2 x+5 x+5-1=0 \\ & x^2+7 x+4=0 \Rightarrow \text { one root satisfying } x<-2 \end{aligned}$$

$$\begin{aligned} & \text { (II) if }-2 \leq x<-1 \\ & -x^2-2 x+5 x+5-1=0 \\ & x^2-3 x-4=0 \Rightarrow \text { not root satisfying }-2 \leq x<-1 \end{aligned}$$

$$\begin{aligned} & \text { (III) if }-1 \leq x<0 \\ & -x^2-2 x-5 x-5-1=0 \\ & x^2+7 x+6=0 \\ & x=-1 \text { is only root satisfying }-1 \leq x<0 \end{aligned}$$

(IV) if $$x \geq 0$$

$$\begin{aligned} & x^2+2 x-5 x-5-1=0 \\ & x^2-3 x-6=0 \end{aligned}$$

one root satisfying $$x \geq 0$$

$$\Rightarrow$$ The number of distinct real roots are three.