Will the AMD FX 8370E work on my motherboard?

I agree with the others. Updating the bios first is a good idea (to make sure it'll support the new chip) and not overclocking is also sound advice. Hell, even stable overclocks can cause instability, throttling, or even reduce performance depending on the quality of your components, your PSU, and sometimes even how a program is written.

Personally I have better performance most of the time when I don't overclock. I blame my cheap Mobo and my crappy Raidmax PSU for that.

Edit: Anyhow, have fun with the 8370E, it's a very good and highly underrated chip, imho. If I could go back and do my build again, I would've gotten that chip for the good performance with a 95w TDP.
 
This board can handle a FX8370E just fine. I run it Overclocked, @ 4.1GHz @ 1,31v or 4.3ghz @ 1,37v if i want extra performance, with a FSB@210 with RAM overclock @ 1959mhz 2x8GB Crucial 1,5V DDR3 @ 1,65v CL12.
NB/HTlink @ 2310mhz minimum or auto, and sometimes OC to 2520mhz. I can prove this, here's my score in CPU-Z http://valid.x86.fr/yjcrmy

Paired with RX470 Nitro+ 4GB @ 1350mhz - Gaming (real world experience) is almost identical with i5-2500k stock performance, in smoothness even thanks to fast ram, i can achieve 144fps with low settings, and my GPU is up to a RX470 4GB @ 1350Mhz which, at 1330+mhz is around R9 390 performance and always hits around 99% usage.
The board supports power draw on CPU line up to over 175w (205 is approx. max) with 80-90c TDP VRM, and has heatsinks+solid caps.
Chances are you will very easily get 4GHz even on a fullfat FX8370 with normal stability, but do not OC then.
I achieved 4,4ghz @ 1,43v so 4,5 even on a PGS batch fx8370e is probably undoable.
Still this is the cheapest am3+ board and most stablest micro-atx released AFAIK even in 2017. Its not fastest but it will do the job.

FX8370E is perfect for this board.