Update 31 dec 2013. A miracle occurred. This morning my Gateway started with Windows 8.1 in 20 seconds or less. I have shut down and restarted twice with this successful start up rate. After the first start up I checked for updates and found none installed. Before restarting a second time, I found about 20 updates and I elected to install these immediately. One update that may have relevance is KB2887595, however, I installed this after the first successful start of 20 seconds. I am now using the "Charm Bar" method of shut down. For new Window 8 users, mouse over to a right corner of the screen, see a vertical menu, click on settings (lowest icon on listing), click on the power options, and then select shutdown. Things have improved significantly over one day. I have no real idea why, except that I used the Charm Bar Last night. What didn't work for me was unplugging all external devices, searching and not finding a bluetooth driver that I could disable. I decided to participate in Microsoft's customer experience programs for the first time yesterday, so that my "experiences" are being sent Microsoft. I was preparing to call Microsoft this morning, further, I was preparing to try reformatting the hard drive, and reinstalling Windows 8.1 Pro again. Neither of these action were required today. The operating system seems to be working very well on my Gateway. One day and counting. Happy New Year.
Early in my debugging, before solution found, I tried many times to use the charms bar to shut down. In no situation did it decrease the shut down time. Fortunately, something different happened to your computer and it is well again! If you discover what the change was, please let us know.