Hi Chris,
Thanks for your interest with Syncfusion products.
By using the following properties of the TabControlAdv we can set the close button for the Active tab.
[Code Snippets]
this.tabControlAdv1.ShowCloseButtonForActiveTabOnly = true;
this.tabControlAdv1.ShowTabCloseButton = true;
And also here I have attached the sample for your reference.
http://www.syncfusion.com/downloads/Support/DirectTrac/88951/TabControlAdv865677741.zipPlease have a look at the above sample and let us know if you have any concerns.
Regards,
Rajasekar R.