Thank you for finding the solution but I have already found it.
I must proof that this equation has only one decision. (I can't proof it using graph because it is enough difficult to construct y=(4+sqrt(15))^X + (4-sqrt(15))^X without, for example, WolframAlpha).
I don't know how to solve this equation either and I don't see how letting will help.
On the other hand if you graph the function
with
you will notice that it is decreasing and that
This shows that is the only solution
To prove that it is the only solution you would have to show