I too purchased a Gateway ZX4250 and a 4 GB RAM upgrade kit. It took me awhile to figure out how to open the case. Here is what I found.
1) Remove the two rubber pads at the bottom of the case (don't worry they pop right back on).
2) Open the small back cover.
3) Remove the one screw.
*****CAREFUL****
4) This part is a little difficult but just be careful not to use too much force. Take two small flat head screwdrivers and slowly separate the plastic silver front panel from the plastic black back panel. Start at the bottom where the two rubber pads were removed. Work your way around the case, you can see how the clips are connecting the two panels as you work your way around the all-in-one. It shouldn't be too hard just be careful on how much force you use.
5) Once the panels are separated you can now access the hard drive, optical drive and some other components.
6) Disconnect the Hard drive sata cable from the motherboard.
7) Disconnect the other cable right next to the sata cable on the motherboard (I'm not sure as to the purpose of this cable.
8) Remove the screws from the metal plate covering the motherboard. Take careful note that a grounding cable is connected to one screw.
9) Install your RAM.
10) Reverse the procedure to put everything back together.
This whole procedure to upgrade my RAM took about 15 minutes. I hope this helps some of you that are trying to upgrade your RAM. If you have any questions PM me.