My Fire HD8 seems to work fine with bluetooth to my Echo Dot. The sound quality is as good as one would expect from a Dot. It sounds better when connected to better speakers (duh!)
I assume you checked the obvious - playing different music files on the Fire to make sure you just don't have one bad file that you ended up testing with? Also, moving your Echo Show and Fire tablet to a different location to check if there was any kind of local interference causing the problem? Do you have a different power adapter that you could plug into the Echo Show as a test - you might have a switching power supply powering the Echo Show that is beginning to fail. Try moving the Fire and the Show farther apart (counter intuitive, I know, but we're trying to see if one may be causing interference to the other). Does the Show have a corded audio input (I don't have one so I don't know)? If so, do you have the appropriate cord to connect your Fire to the Show (therefore bypassing bluetooth)?
What we're trying to do is isolate things. Is it the Fire? Is it the Show? Is it the connection between them (bluetooth)? Is it interference caused by other nearby devices? Is it a bad audio file? Is it a bad power supply?
Just this last Tuesday we were having a roundtable discussion among us ham radio operators, "Have you ever had to deal with interference? Have you ever created interference?" Some of the stories about electronics affecting each other were pretty humorous. One guy could key up his 5 watt handheld (transmit with it) and his computer monitor would turn off. 5 watts isn't much power, and a bluetooth radio is far far less. But you never know. LED lightbulbs are notorious for creating radio interference and noise. If your Fire, Show and an LED table lamp are all right together they might be having this little interference party between themselves that is causing you fits. Throw in a noisy power supply to the Show (maybe you accidentally mowed over the power cord with a vacuum and nicked it) and now you have a four-way noise and interference party going on.
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