It might be interesting to try your idea, to see for yourself where the difficulty lies, but be warned that finding general solution formulas for degree 3 and 4 equations is kind of brutally difficult. I'd link to the historical developments but it would spoil it for those that nevertheless want to give it a shot. Degree 5 requires some serious far out of the box ideas, if you want to understand what's going on.

But in general the idea of making small changes to solved problems and seeing if you can tackle them is an excellent approach to recreational mathematics! I think I'd start somewhere other than higher degree equations though, that would be a rough start.