Simply do the following:
Put a static ID of your region. For this, go to your region property -> Advanced-> Static ID. I have put 'CARUSELCONTENT' as Static ID of my region
2. Now go the Region property of 'Header Text' and type the below command
and save it. Now Run and see the desired background color of region.
3. You can put this Style code in the Page header as well. it will work same. For this, go to Page property -> HTML header
Go to your region property -> Advanced -> Custom Attributes. Now type the command style="background-color:red;". Now save the changes, Run the page and see the region color.