Yes, subject to the fact that notebooks are not designed for 24/7 operation as desktops are -- you will shorten the lifespan of the system if you run it this way.

There are plenty of less expensive ways than a high-end notebook to do this job -- if you're displaying video, even a $200 HTPC will do the job just fine without committing a $2K notebook to the job.

Thanks all ready have the laptop but since I no longer work would save me also getting a desktop since I already have a TV in the home office and Bluetooth keayboard

You really shouldn't have any issues.  The Alienware 17 is a desktop replacement laptop.  If you are going to keep it plugged in though, take out the battery.  

OK thanks but unfortunately the battery does not come out on this modle

Ahh, then I wouldn't worry about it.  I would cycle the battery every now and then by using it unplugged for a bit and then letting in charge.