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I don't think this equation can be solved analytically, but I may be wrong. Here is how I found the solution.
I used the following command to plot the function
plot(x+sqrt((x+1)*(x+2))+sqrt((x+2)*(x+3)+sqrt((x+3)*(x+1)))-4,x,-10,10)
I see that this is zero at two points near -7 and 0. So I typed the following commands to find the root in that region
find_root(x+sqrt((x+1)*(x+2))+sqrt((x+2)*(x+3)+sqrt((x+3)*(x+1)))-4,-1,5)
find_root(x+sqrt((x+1)*(x+2))+sqrt((x+2)*(x+3)+sqrt((x+3)*(x+1)))-4,-10,05)
This gave me -0.06139800265618307 and -7.529020910031573.
