This ram from crucial is probably your best bet. It's CL11 latency so, exactly what HP uses from the factory.
https://www.amazon.com/dp/B0091LG13O/?tag=tonymacx86com-20
I trust Crucial's quality control much more than A-Tech. Crucial does give you a lifetime replacement warranty so that is a bonus.
If you want some heatsink rather than none you can spend 2 dollars more per 16GB kit. Should be slightly faster due to the better CL9 latency. Timings are the same as the G.Skill ram that I use in my HP. 9-9-9-24
https://www.amazon.com/dp/B006YG9EEW/?tag=tonymacx86com-20
BLS2KIT8G3D160ADS1S00 is a 16GB kit consisting of (2) 8GB DDR3 modules that operates at speeds up to 1600 MT/s and has a CL9 latency with extended timings of 9-9-9-24. It is Performance and is non-ECC.
I've never used the Corsair ram in one of these so I'm not 100 % certain those will work or not. Maybe someone else can chime in if they have used that.
Are you going to use HD4000 or a dedicated gfx card in your HP 8300 ? If it is HD4000 then spend the extra 2 dollars for the Ballistic Sport ram. igfx uses some of your systems ram to function. Up to 1.5 GB so faster ram is better.