Basic Inspiron 3668 GPU upgrade to NVIDIA GeForce RTX 3090 Ti 24GB GDDR6X PCI-Express 4.0

you need to upgrade the psu.  stock psu is 240w too weak for the gpu.

how long is the video card?  If it is 336 mm it may be too long to fit the 3668 case.  The case is only 303 mm deep.

motherboard has one 6+4 pin.  you can upgrade to SFX psu with 24 to 6 pin adapter.

The stock power supply in the Inspiron 3668 is only 240 W. NVidia says the GeForce RTX 3090 Ti needs an 850W PSU.

So no, it's not going to work. And unfortunately the PSU in the 3668 is a custom design and not easily replaced.

Re:  hoping to get information regarding any other required/optional upgrades I may have to do in order to get it to work

most 3090ti card would be too long for the pc case.  But there is one more serious drawback.  Suppose you find a psu powerful enough and a card that fit the case, it has such poor air flow that you will be roasting the expensive card in it.  

If you want to stay on intel 6th gen cpu of this pc, salvage the Dell cpu, ram, ssd but forget about the Dell proprietary odd shape motherboard that cannot be moved to new case, get a standard 100 series motherboard such as XPS8900, a standard ATX case that has high air flow (such as one that has a large side panel fan directly blowing cool air on the gpu) and a standard ATX psu.  These parts do not cost much because they are old fashioned and widely available.  It will save you money and major headache than trying to mod the Inspiron case (SFX psu costs much more than ATX).  The Inspiron minitower was not meant to be a gaming platform. It was for common folks surfing net and email or streaming video.  It is totally inadequate to handle a very power avid 450w video card.

PS a standard ATX case + psu would be future proof should you want to upgrade motherboard and cpu to better match the 3090Ti in future.  The only life time limited investment now is standard 100 series motherboard in order to keep using the cpu in Inspiron.