Solving Poisson's equation $∇^2w+1=0$

Question

I'm trying to solve Poisson's equation $∇^2w+1=0$, where $w=0$ on a 2 by 1 rectangle boundary. I just learned Poisson's equation so I'm really not familiar how to do that (I wish I could show what I've tried). A general form of the equation is given as $w_{xx}+w_{yy}+p = 0$, from my notes it seems like we need to first let $w = (p/4)[u-(x^2+y^2)]$, then transform $u = u_1+u_2$ then use the separation of variables. I'm not pretty clear why do we want to do that, and what's the general strategy to solve this kind of differential equation? Many thanks for the help:)