I asked the question How does one attack a divisibility problem like $(a+b)^2 \mid (2a^3+6a^2b+1)$?. I received some excellent comments, one of which has, I believe, given me the idea(s) I need to proceed.
Now I would like to add a bounty on finding solutions, or bounds on solutions, to the divisibility condition in the title. (I currently know of only two.)
Should I add the bounty to the original question (even though the bounty is not specifically for answering it)? Or should I start a new thread — specifically titled so that it's clear that I'm looking for numerical solutions — and eventually attach the bounty there?