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Begin by getting whatever you want to take a screenshot of onto the screen of your Fire Tablet or Fire Phone. This could mean browsing to a particular website or simply getting to the right screen in one of your apps.
Kindle Fire Tablets (3rd generation + higher) and Fire Phones
Press the Volume Down button + Power button at the same time and hold them for a moment. Your screen will flash briefly and your screenshot will be saved in your Photo Gallery.
Kindle Fire (1st + 2nd generation)
As those devices do not have physical volume buttons, there is no easy way to take a screenshot.
You can however simply use another device to take a photo of your screen. Just make sure that the picture isn't blurry or out of focus, and that any details are easily recognizable.