Can this run battlefield 4?

Solution
APU's aren't very good for gaming. As for "fluent gameplay" it totally depends on the game but there aren't many modern games that would run very good on this. BF4 would run like crap on this.

I've seen benchmarks that indicate this APU is okay, but everybody I've talked to says they regretted the purchase.

I strongly recommend getting the X4-750K CPU instead and increase your budget to get and HD7770 or better graphics card.

APU's also share your System RAM (DDR3 memory) so they use up some of that memory and also are bottlenecked if it's not fast enough (i.e. 2133MHz).
APU's aren't very good for gaming. As for "fluent gameplay" it totally depends on the game but there aren't many modern games that would run very good on this. BF4 would run like crap on this.

I've seen benchmarks that indicate this APU is okay, but everybody I've talked to says they regretted the purchase.

I strongly recommend getting the X4-750K CPU instead and increase your budget to get and HD7770 or better graphics card.

APU's also share your System RAM (DDR3 memory) so they use up some of that memory and also are bottlenecked if it's not fast enough (i.e. 2133MHz).
 
Solution
Example $450 build: http://ca.pcpartpicker.com/p/dcMYQ7

1) I recommend adding an identical memory stick later on to get up to 8GB. I didn't go with 2x2GB because there were only two slots.

2) If I drop below a GTX750 graphics card, the performance drops off very quickly. In particular, look at the HD7750 and GT640 relative to the GTX750Ti: http://www.techpowerup.com/reviews/ASUS/GTX_750_Ti_OC/25.html

A GT640 costs about $90 and gets 35% the performance of the 750Ti that costs $150.

*For reference, the GT640 when paired with the A10-5800K (replacing the graphics part) got about 50% improvement so the GTX750 + X4-750K might be as much as 3X the performance for about $50 more for the build!

To repeat, the X4-750K + GTX750 system, would probably average about 3X the frame rate as the A10-5800K setup. (Unfortunately I can't find everything on one benchmark)