Fermi :
Ok so I finally got some time to troubleshoot this laptop. When I turn it on I can hear the drives spool up, and some of the lights above the keyboard come on but that is all. Pulled memory, and cpu and no beeps with either one pulled. My first thought are a bad motherboard, and possible cpu too. So I guess I will be on the hunt for a replacement motherboard, and the graphics will have to wait for another time.
Are you able to boot into windows? get minitool partition wizard and wipe the whole disc, then get a windows installation media and reinstall windows and after the fresh installation flash and update your BIOS. After this process you'll have cleared up the possibilities of bad software. Be careful when buying refurbished/used items, you may have expected to save money but you may even end up with an used and old product, paying more, having a hard time and even screwing up everything. Laptop components are not easy to get, it's not like you'll find an nvidia gtx 880m at amazon. if you change your motherboard you may have to change your CPU/GPU too in case they are soldered or the socket isn't compatible in which case you'll have to buy either one of them or both (are you feeling lucky?). A mobo is around 300$, an Nvidia GPU can cost anywhere between 100$ to 600$ (your best bet is the 860m which costs 200$), an Intel CPU costs between 200$ and 1000$, your best bet is an i7 4700mq for 400$ or something among the lines. If you manage to get everything working without spending much money then I guess you didn't make a bad purchase, but in case you do I can tell you could've spent 1000$ for a new (first owner) product with an 860m, 7200rpm 750GB HDD, 4700hq, 1080p res., no hassle, and probably better speakers, screen, battery, microphone, camera, etc. (judging by this being a product delivered years later after technology has taken an step forward). Remember this is a 2006 (or before) product, If I were you I wouldn't dare spending a penny on it, it's 8 (or more) years old and none of the Hardware I have mentioned should even fit up in your case and let alone work.
If 250$ is/was your max budget you should've bought a 2014 chromebook, it sure is better than an 8 years or even older Alienware
Are you able to return this product? is it under warranty? Never buy used items from untrusted sources without warranty