I suspect it may be a problem with the video cable coming from the convertible tablet, up through the hinged support for the LCD screen, AND/OR the screen Inverter.
I would like you to connect a VGA monitor, to the VGA port on the back of the convertible tablet. The VGA monitor will now be an External Monitor.
[ Looking at the rear of the convertible tablet, start at the Left side.
Vent, USB port, Monitor port. The Monitor port is the VGA port ]
The VGA monitor can be a CRT type, (Looks like a small TV), or a flat LCD screen monitor with a VGA cable.
Notebook OFF, monitor OFF, connect the VGA monitor to the M275's VGA connector. (Port)
Turn the monitor on. If a CRT type, allow it to warm up.
Turn the notebook on.
By the factory default settings, whatever is shown on the convertible tablet's internal monitor, should now be shown on the external monitor.
If not, press, and hold down on the Fn key (On the bottom row of the Keyboard, all the way to the Left), and tap once on the F3 key.
(Top row of Function keys. The F3 key is the Display Toggle-Over Key )
No graphics shown on the external monitor? Hold down on the Fn key again, and tap the F3 key again.
There are at least three options available;
1) Convertible tablet's internal monitor - ONLY
2) Convertible tablet, AND external monitor
3) External monitor - ONLY
If graphics are fine on the external monitor, you have a problem with the video cable, and/or the screen Inverter. At this time we will bypass the LCD screen, and come back to it.
At the bottom/middle of the tablet's LCD screen, is the Hinge. The Hinge Cover is 'T shaped', and has one single Philips screw holding it on.
If you remove the screw, and the Hinge Cover, you will see two wire harnesses.
One Black, the other White.
See that these two cables are plugged in tightly to the connector.
After checking this step come back in a Comment, and state your findings.
Here is the Service Manual (Free) for the Gateway M275 Convertible Tablet,http://www.tim.id.au/blog/tims-laptop-service-manuals/#toc-gateway
Where it states - Gateway 400, go down to the second line, and across to M275. Left-click on M275.
This is a PDF file. It may take up to 30 seconds before the first page comes up.