Good, so the graphic processor is the HD 3000 that can definetly do what you need, and since you know how to use
drivers, install the newest one as I don't think it's a
driver issue, at the moment.
So we know what laptop it is, as
Medion 6228 exists (that's the support page where you find drivers), the "Medion 6882" like you said in the OP does not.
Let's start: power up the laptop
without the HDMI cable connected, when it is ready to use, attach the HDMI cable.
Go in the Screen Resolution, and look at it, do you see two screens in its window? (one with a big
1 and one with a big
2, like
this) If not click on Detect, and the second will appear in that window.
Then click on Identify. You should see a big 1 on the laptop's screen and a big 2 on your TV.
If this does not happen or if later on you experience sluggishness or flickers or whatever, skip down to part where I talk of BIOS options (bolded for convenience), then retry.
Now that you have two screens in that window, change the first entry (Display) to select the TV, and set its resolution (second entry) to "native" or "Recommended", or maximum.
The last entry in the Screen Resolution window (regardless of what screen properties you are looking at) should be the Multiple Displays, and playing with that you can:
-use only laptop screen (usually the first screen)
-use only connected screen
-extend laptop screen into connected screen
-the reverse of the above (useless imho)
-show the same things on both screens (and depending on what is the main display, will make the other look bad)
There is also "make this my main display" option when you select the secondary display in the first entry, ticking that will move taskbar to the external display among other things as told above. (there are very good third-party multiscreen softwares that allow you to have taskbar on both)
If that works, you can set the TV screen as main screen, and leave the HDMI connected when the thing is powered up. It should output directly to TV if connected without you doing this fuss every time you connect it (as it will remember the device and the relevant settings), and use laptop screen if not connected.
Looking in the
BIOS options at how much of your RAM it can use (as any integrated graphics processor it has no ram of its own, and steals some of your RAM to do its job), raise the allocated RAM to the max (normally it's set at 64 mb, you should be able to increase it to at least 512 mb).
For accessing BIOS options, look at what it says in the first (text-only or with a non-windows logo) screens when you turn on. It should tell a key to press, it's usually Del or F2. Then you navigate inside the BIOS tabs and entries with arrows until you find Graphics Shared Memory or somesuch and change the values.
This should decrease the amount of RAM available to the computer by the same amount, for now it's just to make sure that it isn't causing issues, if it works you can dial that down to 256 which is the max it should realistically be able to use, or less. You have 4 GB of ram so it will run fine for these tests (and even for general use for that matter) even if you leave it at higher dedicated RAM.
As for BIOS version, that could be another possible solution, but I don't see any BIOS update in the support page I linked above.
There are sometimes issues about TVs not able to be set the right resolution, but that's a problem that could come after your current one (as you don't even manage to use the second screen other than seeing boot sequence), so
here is the relevant page.
Try and report back.