You guys gotta tell me...

Mute them, encourage them to mute each other and stick to pings. At least try not to get negative.

I don't play anymore unless I have one friend on at lease cause then we can try to two man it. If it gets really bad we will roam jg and get any ganks we can. Playing solo it is really hard to enjoy the game anymore even in the fun game modes like ARAM and seasonal game modes people just get way to mad.

If they won't stop just write the game off as a loss and don't immediately jump back into queue (also report where necessary). Watch an episode (or two!) of your favorite show and then go back into queue. That way at least in your next game you won't have to deal with those four idiots. If you keep getting matched up with people who dissolve into childish squabbles then its time to quit ranked for the day and play something a little more lighthearted. You can then tackle SoloQ when you've given yourself a reprieve from the frustrations it can bring (and bring back your focused mindset).

It's maddening I know but sometimes you can do absolutely nothing about it. You can also lessen the amount of unnecessary friction by bringing in someone you regularly play with. Listening to music (it doesn't matter what kind as long as you like it) can help you keep your head in these situations too.

You don't really NEED to stay positive. You should be safe from reports/bans as long as you don't become NEGATIVE.

It essentially just boils down to muting them and gritting your teeth. Maybe take a break afterwards to vent and come back ready to play the next, hopefully better match.