Let \(x,y,z\) be positive real numbers such that \((x \cdot y) + z = (x + z) \cdot (y + z)\). What is the maximum possible value of \(xyz\)?

see correction below

I don't think this is correct

You're right. Not sure what happened.

Again using Lagrange multipliers I'm seeing the solution as

x = y = z = 1/3

xyz = 1/27