Whenback in September, both were equipped with a newly updated (and newly monikered) operating system: Fire OS 3.0. But plans were already in the works for a 3.1 update that would add even more functionality.
That update has arrived, and Amazon plans to deliver it automatically, over the air, to compatible Kindles in the coming weeks. However, if you'd rather get it right now, you can.
Step one: Make sure you have a compatible Kindle, and make sure it's at least half charged. The OS 3.1 update is available for the second-generation Kindle Fire HD, the Kindle Fire HDX, and the Kindle Fire HDX 8.9.
Step two: On your PC, head to the Kindle Software Updates page, then click the link for your particular model.
Step three: Scroll down a bit and click Download Software Update 13.3.1. Make sure to take note of where you're saving that download. (Typically it'll land in a Downloads folder, but you may prefer to save it to, say, the Desktop for quicker access.)
Step four: Using your USB sync cable, connect your Kindle Fire to your PC. Wait until it's detected, then open up File Explorer and click Kindle Drive. Navigate your way to the Internal Storage folder.
Step five: Locate the software update you downloaded (it should have a .bin extension), then drag it to that Internal Storage folder. (Make sure you don't copy it to a subfolder, of which there are many.) After the file has been copied, which should take no more than a minute or two, you can disconnect your Kindle from your PC.
Step six: On your Kindle, swipe down from the top of the screen, then tap Settings. Tap Device, then System Updates. Finally, tap Update. Wait a few minutes while the tablet reboots and installs the new OS.
That's it! Now you can enjoy the various fruits of Fire OS 3.1, including Goodreads integration, Second Screen viewing on select Samsung TVs and the PlayStation 3, mobile printing, and 1-Tap Archive to free up storage.