Ram compatibility comes down to what model of board the system has in it.
The bios firmware version the board currently has.
The amount of slots provided on the board for memory modules.
And the type of memory controller used with the chip set version of the motherboard.
You will find that if the system you have was pre built by a company such as dell for example, or acer as a ready to run out of the box system with a pre installed os.
That the motherboard used is custom made for the model version of the system.
Often they have a very basic bios and firmware.
meaning that when you enter the bios you have very limited options and features that in a retail motherboard you will have.
Options such as the speed ratings of the memory and settings for it are limited.
Leaving you with a very small amount of choice when it comes to the speed and capacity of memory you can buy and will work with the system.
Your board will support a maximum of 16Gb of memory.
That means each of the four memory slots of the M3970g will support a maximum density size of memory of 4Gb per memory stick or module.
Please click on the link bellow to see what memory kits will work with your M3970g system.
Based on the brand name you can buy the memory stated as working, or known to work with the system from any place you choose that sells memory for upgrading the system.
http://www.crucial.com/usa/en/compatible-upgrade-for/Acer/aspire-m3970g-uw10p