Putnam 2005 B5

Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the variables $x_1, \dots, x_n$, and suppose that $$\left( \frac{\partial^2}{\partial x_1^2} + \cdots + \frac{\partial^2}{\partial x_n^2}\right) P(x_1, \dots,x_n) = 0 \text{(identically)}$$ and that $$x_1^2 + \cdots + x_n^2 \text{ divides } P(x_1, \dots, x_n).$$ Show that $P=0$ identically.