The computer should be able to handle WoW. However, the questionable point is the video card. The hd 3200 is currently a low-end card, so you may not have good results with it. You may want to buy an upgrade to something newer.
This list shows how low the HD 3200 is in relation to new cards. My prior vid card was a Nvidia 7300 GS, which was 2 steps below the HD3200. WoW ran ok, but the video settings had to be turned pretty much all the way down. Once you're in Northrend in general, and Dalaran in particular, the game really lags with low end hardware. I currently have a HD 4830, and the game runs super smooth now, with all settings on high. That card model has been discontinued, but you can still pick up good versions of it at NewEgg for less than $100. Check out the other pages on the link above for a quick run down of current video card options and price points.
Just bought a Gateway DX 4200 at a great price, was curious if it could run WoW decently beyond minimum settings:
4 GB DDR2 SDRam
Ati Radeon HD 3200 (onboard)
Quad Core AMD
Been playing on a Laptop and it has been a miserable experience, I'm not looking for blow me away Top End graphics here, just something I can play with better FPS and less fog, thanks!
I also have a Dx-4200 desktop, and have to tell you that when i went to upgrade I ran into alot of headache. Not only will you have issues with the graphics card fitting in the small and tight case they give you with it. The drivers for my first upgrade (XFX 4850) did not want to install at all. I later purchased an invidia card that installed better on the amd motherboard?!?.Even now with 6 g of ram, 650 power supply, and nvidia gtx 260 i still cant get WOW in the 80 fps range and still drag to fps as low 20 in daralan. I am finding out now that the phenom 4 9100e that came with it only runs at 1.8, and for gaming that is shitty(great for multitasking though). I am down grading to a faster duel core for gaming hopefully this will help. By the way you will probably have get out a pair of snips and tear away at the case if you ever decide to put a more high performance video card in, I did. But if you do decide to go threw all this trouble like i did, be sure and turn the onbaord video card off in the bios....best of luck to you!