Your discrete graphics chip may have failed.
Install gfxCardStatus https://gfx.io and set it to "i" (integrated graphics only). It will allow your MBP to run on integrated graphics only, bypassing the discrete GPU which has the issues. This may allow your MBP to run normally, although it will have reduced graphics performance when performing demanding graphics tasks. The alternative is replacing the logic board, which is not cost effective on a machine that old, unless gfx does not resolve the issue and you really want to keep this MBP.
There is an acknowledged bug in the current version of Cody Kreiger's Open-Source gfxcardstatus, and the developer has confessed he does not have time to fix it right now.
There is a fork off the main build by steveschow available that seems to fix that problem for current versions of MacOS such as ElCapitan and Sierra. He provides a finished .app for direct download -- you do not have to compile anything.
https://github.com/steveschow/gfxCardStatus/releases
Also note that if your Mac does not run long enough to allow gfxcardstatus to be added, this is not really practical.
In addition, Steve Schow writes that he has abandoned further development -- because there are better solutions available [for both the 2010 model and 2011 models]. In particular, the use of ArchLinux bootable CD to gain access to and re-write the EFI on the drive, and permanently disable the discrete graphics chip. This page and scroll down past the list to the blog:
https://github.com/steveschow/gfxCardStatus
There are two similar procedure listed. I used the second from MacRumors as it seemed easier. I have made the Arch Linux bootable CD on another Mac, and tried this approach. I now have a perfectly-functioning MacBook Pro late 2011 15-in model with Discrete Graphics disabled. Runs just fine. But NO external display support any more.
