Certified GKH Gardening Expert
Well, I don't know about forever, but you can overwinter begonias and impatiens root quite easily so it's possible to propagate new plants each year for a continuous supply. These articles may help: https://www.gardeningknowhow.com/ornamental/flowers/begonia/wintering-begonias-overwintering-a-begonia-in-cold-climates.htm
I have kept impatients in my house over winter but they don't do very well. However, I have several plants that did ok and are blooming beautifully outside.