find the maximum value of
This is a lengthy case of numerical calculations
All you have to do is complete the square in the original, solve for x or y, imput it into the second equation, then differentiate it, solve for zero, and then test the results in the second derivative to make sure they are maxs or mins. If there turns out to be more than one relative, just plug in the values into the initial equation and the largest value is the absolute max.
If someone would care to take the time to do this problem they can have all the credit.
I'd say to use a Lagrange multiplier, rather than brute force. As for doing it without calculus altogether, I'm not sure how you'd do that, but I imagine the fact that the constraint is the circle with center (2,3) and radius (and thus passes through the origin) may be of some use.