Hello,
From what I understand, your game attempts to launch (showing the windows loading icon as the cursor) then shuts down with an error.
This could be caused by having the wrong version of Visual C++ installed, the game depends on this to run correctly.
This will not be the issue on Windows 8/8.1.
If you run Windows 7 or below, please try installing this:
http://www.microsoft.com/en-us/download/details.aspx?id=40784
If you run Windows 8/8.1
This was mentioned in this developer post:
http://help.ea.com/en/article/missing-msvcp120-dll-at-the-sims-4-startup/
Please also read that and see if it can solve your issues.
For some people unable to start the game, your graphics drivers may be outdated. Outdated graphics drivers can and do cause games to not start.
Hope this helps!