Re: A few questions regarding a Roku Express (HD, not 4K) and the Peacock app
I can't offer answers specific to Peacock, but I can offer some general knowledge.
No unless there's a Wi-Fi issue. If the living room is significantly farther from your router or there's more walls between it and your router then there could be a Wi-Fi issue. Also, make sure you use the supplied power adapter and not the TV's USB port to prevent a problem with the TV's USB port not providing enough power.
I can't speak for Peacock, but usually these channels/apps limit the number of simultaneous users on the same account. If you stop using Peacock at one location and start using it at another it won't be a problem.
The PS4 issue is a little concerning. The only thing that comes to mind is that the TV is so old that it doesn't support HDCP (copy protection).
Just try it out before your friends arrive.
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