First try Oxyclean purchased from any store. Then try 1Tbs ammonia in a cup of water and scrub softly with a brush.
Ice works sometimes, to solidify the gum, and let it to be softly scraped off with a knife. Leave the ice sitting on the gum long enough to freeze it hard, then scrape the gum off.
Starting a cleaning business does not need large savings, or much expertise. However, running a successful carpet cleaning business needs some knowledge. Also, you'll need to have the essential equipment to complete the job.
Stretch it! Buy a knee kicker or power stretcher and stretch it out
I love OXI-clean - I've removed even red wine stains with it! Try it. It works great for me.
There is a product called OOPS that might take away, if it is Latex and not huge. Go to the hardware store and inquire in the paint department.
Better confirm what it will do to the carpet before you put it on a visible place.