Anyone would be yelling and swearing after dealing with HP in my own personal experience.
If the screaming is definately coming from your speakers - the screaming is most likely feedback between your speakers and perhaps a microphone on your monitor. Disconnect the speakers to begin with. Start your PC and turn off the boost on your microphone advanced settings and reduce the volume of your microphone and speakers. Plug your speakers back in and adjust the volume until you have a comfortable compromise. Mute the microphone when you are not using it to enable higher speaker volume.
If it is at all possible that the screaming is coming from the computer itself then I would check all your fans as these can make a real ugly loud noise if they are not spinning correctly or if they are not spinning together in sinc. A bit of sewing machine oil on the bearings of a noisy fan or a replacement fan may be required. A fan not working may also produce a loud noise and may also be the reason you have had a non starting PC also.
Lack of fan activity can result in heat - if your computer simply doesn't start up anymore, then you need to check your vitals - Processor, HDD and RAM to ensure that they are not heat damaged.
I assume since HP have looked at it 4 times that it is still under warranty. If not and you paid them to look at it, check what warranty comes with the repair as the same fault should be covered by some kind of repair warranty.
Other than this I would just say provide more details about what you were doing when it first started screaming, what system you have, what software and what additions you have made from new. All these things will help others to better diagnose your problem and hopefully find a solution.