What to do with long equation-heavy solutions?

This happens not just in mathematics but in many other disciplines as well: modeling of biological systems, for example, often involves a vast number of terms in the equations. The best approach that I have found for these sorts of situations is to look for the structure in the system and to build your presentation around that structure.

In most cases that I have encountered, it is not true that there are simply a bazillion essentially unrelated terms. The true story of the equation, then, is not the expanded mathematical form but the process and relations that generate it. This opens up a number of approaches for factoring out the equations in order to make them more tractable to present, including:

• Abstracting sub-structures as variables (e.g., X^2+XY+Y^2, where X and Y are complex terms presented separately)
• Separating variables from parameters (e.g., a reaction network where each reaction uses one of several standard Hill equation models, with the parameters of the models presented in a table)

These sorts of factorings also lead to a much more informative and interesting presentation of the equation as well. In fact, the Big Equation itself may often end up being relegated to an appendix, where its page-busting form is of little concern.

There may be, of course, certain systems in which such factoring is impossible---and that fact is quite interesting and worth clear discussion!

Quite often, following @jakebeal, a formula with many terms can be factored, or split into groups with similar interpretation: terms with similar powers, or groupings that are consistent (for instance in convective and diffusive terms with differential equations). You may put the emphasis on the most important terms, and group the negligible ones. If you are a LaTeX user, you may use different font sizes: How can I change the font size in math equations? to help the reader focus on important parts.

Or if your equation is parametric, and your paper only requires a subset of solutions, those can be given in the text, the rest can be, as suggested, put aside in an appendix.

If you want montruous examples, you can have a look at What is the longest equation known?