Let $f$ be a polynomial of degree at most $2$ in the variables $x,y,z$. What is the most general such polynomial that satisfies the Laplace equation? How do you find it?
Note: The partial differential equation $f_{xx} + f_{yy} + f_{zz} = 0$ for functions on $\mathbb{R}^3$ is called the Laplace equation.
Since any coefficients of $xy,yz,zx,$ or $1$ are going to be $0$ in the second partial derivative, all that matters in the laplace equation are the coefficients of $x^2,y^2,z^2$. In other words, they must add up to $0$.
So $f=ax^2+by^2-(a+b)z^2+cxy+dyz+ezx+g$ where $a,b,c,d,e,g$ are any real numbers.