One solution you could try for your carpet cleaning issue is renting or buying a carpet cleaner machine. You can use any cleaning solution you prefer with it, and it might solve the problem. It's possible that the issue isn't the solution you're using, but rather the way you're applying and cleaning with it.

Even a less expensive carpet cleaner can work wonders, so don't worry about spending too much. It's worth considering getting one so you can clean your carpets a few times a year. You'll be amazed at how dirty your carpets were once you see the water that's sucked back up.

Plus, since you have white carpets, investing in one of these machines is probably a good idea.