if I look at Dell support website, then there I can see a dozen different optical drives this Inspiron 8600 could be equipped with.
It is not impossible, that one of these drives could be crossflashed from OEM to retail firmware with DL capability as additional benefit.
Dell even had published firmware updates, that officially enabled DL capability:
Sony DW-D56A Slim 8x DVD+/-RW
Release Date: 7/12/2005
Version: PDS7 (Win), A04
The firmware also allows the drive to write to the latest dual layer DVD+R media.
NEC ND-5100A Slim 4X DVD+RW
Release Date: 2/25/2005
Version: 10FE(WIN), A01-1
Adds Read Support for Recordable Double Layer DVD+R
NEC ND-6500A Slim 8X DVD+/-RW
Release Date: 5/10/2005
Version: 203D, A01
Firmware upgrade to support writing to the latest dual layer DVD+R media and various other brands of DVDR/RW and CDR/RW media.