How do I...connect to remote data base through visual studio 2010

There are a lot of possible problems, starting with "is the DB configured to accept remote connections?" - many aren't for security reasons - and ending with "are your credentials correct?". Unfortunately, the error messages generally aren't very helpful (sometimes I suspect deliberately so, so prevent giving hackers information on what they are doing wrong).

Start by trying a connection directly in VS with the Server Explorer pane:
1) Open Server Explorer.
2) Right click "Data connections" and select "Add connection"
3) In the dialog that follows, select your DataSource, and database, specify the security info, and press the "Test connection" button.
4) When the connection works, press "OK"
5) Highlight your database in the Server Explorer pane, and look at the Properties pane. A working example of the connection string will be shown, which you can copy and paste into your app or config file.

If the test connection fails, you need to look at the error message closely - it may give you a clue as to what the problem might be. But without that, we are all working in the dark!