So in Chapter 1 of the FEniCS book section 1.1.3 as relates to Poisson’s problem u_e(x,y)=1+X^2+2y^2.
It’s not that I am refuting it I would maybe like to know a little bit more about how this derivation is formed or if there is another place, book, link etc. to define the details of how such an equation for the x, y values of u_e was came about or formed it?
Usually, you chose a u(x,y) that satisfies a set of boundary conditions, and then you manufacture corresponding source term f such that -\Delta u=f.
This is called the method of manufactured solutions.
See for instance: Error control: Computing convergence rates — FEniCSx tutorial
In addition to the previous answer, I would like to add that if you are interested in this method, you can find more information (including guidelines for creating manufactured solutions) in the report by K. Salari and P. Knupp Code Verification by the Method of Manufactured Solutions, which is publicly available here. For more information on verification in general, see the book by P. J. Roache Verification and Validation in Computational Science and Engineering (Hermosa Publishers, 1998).