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Using Lagrange multipliers, make a new function:

$f = xy^2z^2 + \lambda (x+y+z-12)\,$

Take the partials of $f\,$ with respect to $x\,$ , $y\,$ , and $z\,$ and set them equal to zero.

1) $\frac{ \partial f}{\partial x} = y^2z^2+ \lambda = 0\,$

2) $\frac{ \partial f}{\partial y} = 2xyz^2 + \lambda = 0\,$

3) $\frac{ \partial f}{\partial z} = 2xy^2z + \lambda = 0\,$

This is a trick. Notice that if you multiply the last three equations by $x\,$ , $y\,$ , and $z\,$ respectively and then add them, you can factor out a $(x+y+z)\,$ term that is equal to $12\,$ . So multiplying,