(4, 4, -5)
(4, -5, 4)
(-5, 4, 4)
I saw your question.
I thought that I could answer it.
I assumed, since the claim was that there were exactly three more solutions, and since the equation is symmetric in x/y/z, that the solutions would be symmetric.
I assumed that two of the variables were equal.
I tried a few smaller numbers, and considered powers of small numbers.
I came across the given solution.
I typed in into the box, and hit "Post quick reply".