Question

Solve in real numbers the system of equations: x^3+3y^2+1=4z, y^3+3z^2+1=4x, z^3+3x^2+1=4y

Answer 1

May not give you all the solutions, but as a start I think you may try using the symmetry, \(x=y=z\) :
\[x^3+3x^2+1=4x\]

Answer 2

Notice that if \(x=a\) is a solution to above equation, then \((a,a,a)\) is a solution to the given system