I've gone through many Rex Begonias over the years; they'd seem to do great for a few weeks and then suddenly look blah and die. Even with our high humidity here in Fla. I had trouble keeping them alive for long. Last year I decided to try again and sat the pots on pebble trays and wow, what a difference. Mine still don't look fantastic but that may be because I let them get real dry before remembering to water but with the pebble trays of water they seem to be doing so much better. If your "poopy" one gets to the point where it disappears and seems to have bit the dust, don't throw it out .... the rhizome may very well re-sprout at some point. I had one that I thought was dead so I stuck a cutting of something else in the pot and a few weeks ago I noticed a Rex Begonia sprouting in that pot! These aren't the best pictures because I'm in a hurry and running late for an appointment but I thought I'd share and say thank you for your post because it made me realize I need to water mine. The third photo is the one on my back porch that showed back up after I'd put a cutting of an Aralia in the soil.
~ I'm an old gal who still loves playing in the dirt!
~ Playing in the dirt is my therapy ... and I'm in therapy a lot!