When you try to pass bills in Homeroom, you have a chance of your bill being rejected. When this happens, you are given the chance to "Persuade by Force". If you do, combat will begin, and you must eliminate all reps that voted Nay. You can either kill them in combat, or throw them into a higher-level Yay-voting rep. When thrown into another rep, they will merge into whichever one had a higher level, adding their levels together (up to 9999).
Be careful not to hang out close to Yay-sayers at the end of your turn. If any attack damages a Yay-sayer, they will turn hostile, requiring you to deal with them as well.
Keep in mind that if you put characters into the Election Committee, there is a chance they'll vote Nay, forcing you to take them out.
Once you have eliminated the last enemy, the bill will be passed, and the trophy will pop.
Do not make a habit of using Persuade by Force. Each time you kill a rep, they hate you more and more. Once you've completed a Persuade by Force, focus on buttering up the reps as much as possible, one club at a time. Keep junk items you've collected in the campaign or Item World runs as bribe material. Try to bring a full set of 32 items to give away. Their disposition will eventually become better and better as the game goes on, making bills easier and easier to pass.