Which realtek NIC Driver update?: HP or Microsoft??  

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You could try either separately however be sure to set (create) a system restore point before installing either of the two. In this way if things do not work out, you could always roll back to your current state.
My recommendation would be to go with the HP one first.
BTW, are you having any issues at present with your NIC?

Regards

well, you could split the difference and download the driver directly from realtek. otherwise go with the hp driver. most likely, they will all work. in windows update you can choose to hide certain updates so you wont be nagged every time you run it.

If its Realtek hardware, goto the Realtek site. The mobo manufacturer should also have the drivers at their site. Does HP make mobos? If not, then they assembled the puter from other manufacturers equipment. Make sure to back up and create a restore point before attempting updates. Some updates like to be done in SAFE MODE, some updates want the total removal of existing driver. ie video card

ps: you can also hide MS updates so they do not keep coming up again and again. You should not update drivers in any automatic format! If you need to update, get your hardware info and go find the manufacturers recommended drivers. Be very deliberate about process.
I have W7 auto update disabled, I update what I want, when I want, and how I do it, when I decide. Only time I use windows stock drivers is at initial OS install and for the plethora of little stuff that I do not know any better.