When real-estate agents who work for real-estate companies find homes to sell, what is that process called?

And can the real-estate agent buy one of the found homes for himself/herself?
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Maybe the process you are talking about is prospecting.
I set up some sort of systematic way of finding properties that the owners want to sell and talk those owners into listing them with me for sale- that would be what I call prospecting.

Yes the Real Estate agent or Broker can buy one of these homes themselves. I would imagine a trial court would decide that an agent can not buy a house for themselves and still pretend they are somehow representing the best interest of the owner- but with the proper disclosure the owner could still sell the house to the agent.

I had a listing where the elderly resident could not remember that her home was for sale and wondered how come people would show up. I met with the lady and her grown children and I agreed to buy the house at a slightly lower price and help the lady move on.
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When they look for a home for you they are a buyers agent. If they are selling a home for you, they are your listing agent. There's no real name for either process.

Yes, but laws vary from state to state. They can represent themselves in most cases,but have to disclose that they are a Realtor.