Alienware 17 (2014 model) memory issues

My laptop stopped working about a week ago. The lights went on but nothing was displayed, no beeps and then it went down. I opened the bottom panel and played with memory in both visible slots. I found out that one of the modules is not working in both dimm3 and dimm4 slots and when that module was removed - laptop started normally. Since I removed the faulty memory and tried to use it again - laptop started beeping 4 times - as normally it should do for memory issue. So I ordered replacement memory and on a day it arrived (two days later), before I even tried to put it in, laptop started behaving exactly as above (lights on, no beeps then power down). I emptied both dimm3 and dimm4 slots - no improvement. Then I removed disks, opened laptop fully and removed memory modules from dimm1 and dimm2 slots. Then I put it all back together and started playing with inserting single memory at a time to dimm3/dimm4 slots. I learned that all three remaining modules are now failing. I put the new replacement memory in - and it works fine now. It is like that for 2-3 days now.

To summarize - all four original memory modules are not working properly now (four  4GB 1Rx8 PC3L - 12800S - 11 - 12 - B4). The replacement one I ordered with exactly same numbers on it however when arrived its last numbers were '13 - B4'.

I am thinking that this issue might be caused by motherboard problem but I would rather avoid replacing motherboard as it is too expensive. Any other ideas what happened and how to avoid it? I am now running on only 4GB memory but I hesitate to order  more modules if there is a risk for them to start failing as well.