KVM over IP would allow you to control the computer from another room given that are able to run a dedicated Ethernet connection between the two rooms. There are some rated for 4k@60Hz which would allow you to get some lightweight gaming.
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will this allow the Office terminal to run windows
sorry, I am trying to explain this best that I can.
Maybe a "dumb terminal" is better?
I walk into office , turn on monitor, and then "login" Windows but I have no computer connected in this room but is in the closet
When in the living room, i would use keyboard and touch pad to select a movie but computer is in the closet.
KVM over IP would allow you to use your computer as if it was in the same room as you, if you want to control the computer from multiple rooms you'll want to find what is referred to as a one-to-many KVM, meaning one source to many receivers.
Something like:
https://www.orei.com/products/hdmi-extender-over-cat6-7-with-kvm-hdmi-loop-out-1080p-60hz-up-to-330-ft-ex-330-kvm
Using terminals, thin clients, or low powered machines at the end points would require setting up Windows RDP or running some level of virtualization with Microsoft Hyper-V either of which would require Windows Server to operate effectively. Last I checked Windows Hyper-V does no support GPU pass through.
It sounds like what you want it some sort of HDMI over CAT6 cable with USB over the same cable as well.
I know you can push HDMI (or displayport) a very long way over CAT6 cable with the correct converters - I'm not sure if any of those solutions include remote USB as well - you'd think some do.
Any other solution means you need other computers at each endpoint.