I hate to be the Bearer of bad news, however, I had an Alienware Area 51m 5620 that I bought brand new about 2 1/2 years ago. I had the 2 year warrenty. One week after my warrenty expired, I had those very same lines on my system. It ended up being a bad video chip on the motherboard. THe problem is you cannot swap just that. It is a whole new motherboard. The place I had look at it quoted me 1200 to fix it. Now I knew about this problem about 10 months prior. That particular model had a problem with the onboard video going bad. I hate to say it, but your video is bad on it. Mine had the ATI Radion 9000 128mb version on it. I truely hope you are able to solve, but from my experience, you need a new motherboard.
Many have asked me why in the hell I bought another Alienware. It is simple. My job requires I travel. When I am on the road, I like to have a portable computer capable of playing the same games that my home system does. I dont use the batteries very often so that is not an issue for me. I love my 7700. Plus you can get to everything on this one. If my video goes bad, I can take it out, order a new one and stick it in and I am good to go.