10 points for the best answer. Help me, please!!?

There should be a much easier way to do this, but I'm having problems thinking of one.

If permitted, I would use a mathematical software e.g. maple, or use Excel with a maths plug-in.

If not, you want to do the following:

P = [(y^2+2z^2+3)(z^2+2x^2+3) + (x^2+2y^2+3)(z^2+2x^2+3) + (x^2+2y^2+3)(y^2+2z^2+3)]/[(x^2+2y^2+3)(y^2+2z^2+3)(z^2+2x^2+3)]

Then find all the factors labelled xyz, substitute their value for 1.

Cut it all down until it becomes much more manageable.

You should then be able to find a function of P.

Then, there should be a limited number of points where all variables are positive and yet maxes out P.

I would have recommended graph paper, but it's not very easy to draw things accurately in 3D on a 2D plane

Hope this helps