how to apply laplacian

Question

Basically I want to understand the solution,I am confused here how to apply laplacian $\Delta={\partial^2\over\partial x^2}+{\partial^2\over\partial y^2}$ on the object $M=|f|^2+|g|^2$ where $M$ is a constant and $f,g$ are holomorphic function, I assumed $f(z)=u(x,y)+iv(x,y),g(z)=p(x,y)+iq(x,y)$ so $M=u^2(x,y)+v^2(x,y)+p^2(x,y)+q^2(x,y)$

Answer 1

Hint: $\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = 0$, and similarly for $v$, $p$, $q$.