I've tried dividing both sides with square root of (x-1)(x+3) but I've gotten something very complex. I guess there is some other trick I don't know about.
Can you follow this link?
It suggest that you first subtract \(\sqrt{(x-1)(x+3)}\) from both sides and square the whole equation.
I'd added more about this somewhere, but it seems to have disappeared.View attachment 23276
I observe that, if we call the first two radicals u and v, the LHS is u + v + 2uv. What is the RHS equal to in terms of u and v? A little work, and you'll find that the whole equation can be reduced to an equation in x = u + v. There are probably other useful tricks, but this worked for me. (The solution turns out to be very simple!)