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How many different triangles have side lengths $x,y,z$ that satisfy $3x^3-yz^2 = z^3+4x^2-y$?

I was wondering about this and was wondering in general are there ways to solve such a question for $f(x,y,z) = g(x,y,z)$. In the case of the question above, it seems that there aren't any positive integer solutions, but how do I check for sure and use that to solve the question?

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  • $\begingroup$ There is a positive integer solution, though it doesn't give a triangle. $\endgroup$ – Macavity Apr 26 '16 at 18:27
  • $\begingroup$ Any special restriction on what kind of triangle, or the numbers involved (e.g., integers? rational? real?)? $\endgroup$ – Kieren MacMillan Aug 1 '16 at 20:24

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