A new keyboard is $20-$50 from what I can find on eBay just quickly Googling your laptop.
Possible scenarios I have encountered in the past:
# spill or other damage : the solution is to replace the keyboard here. Find the disassembly instructions probably here on iFixit. It should be simple.
# battery swollen, pinching flat-flex cable : this is really more of a MacBook Pro issue but sometimes the built in lithium could swell causing the flat-flex cable to become disconnected. This could also happen due to some rough handling of the laptop or heavy handed typing flexing the keyboard itself.[br]
[br]
The solution here would also be to attempt a disassembly, then carefully check the connections for the flat-flex cable leading to the keyboard.
Hope this helps![br]
[br]
If you put any more info on here I can probably help more.[br]
[br]
Here are some links to disassembly videos. I can’t actually see a link on the Alienware iFixit page: https://www.google.ca/search?q=Alienware+M17x+P01E+disassembly&rlz=1C1GGRV_enCA807CA807&oq=Alienware+M17x+P01E+disassembly&aqs=chrome..69i57.2063j0j7&sourceid=chrome&ie=UTF-8