15 R4, ethernet not working

Windows 10 should automatically find your device driver and install it. It's possible that it can't auto-install or maybe it has been disabled.

Just check that you have it enabled. Right click on the network icon at the bottom right of your screen. Click on Open Network and Internet Settings. Click on Change Adapter Options. You should be able to see here if your ethernet is enabled or disabled. 

For the driver, go to the Support Page on this site and enter your Service Tag (found on a sticker on the bottom of your machine)

Go to Drivers and Downloads and search for Network Drivers and your ethernet driver should be there. Try downloading and installing this driver. 

EDIT: This should be your ethernet driver but you can check this by using your Service Tag

https://www.dell.com/support/home/en-us/drivers/driversdetails?driverid=p1jp0&oscode=wt64a&productcode=alienware-15-r4

Something isn't right as it says Network Cable Unplugged. Try selecting Diagnose and see if it can fix itself. 

Have a read through this guide and see if there's something that can help your situation.

https://thegeekpage.com/fix-windows-10-network-cable-unplugged-error/

I'd start with repair and see how it goes. It sounds like it may not be a driver issue though. 

Was the ethernet adapter enabled when you looked for it in the Adapter Options?

The first solution didn't work, so I'm going to try the second one "Disable Ethernet Connection and Perform a Hard Reboot". I can't remove the battery on my laptop, will the hard reboot still work?

You can unplug your battery but you'll just have to remove the bottom panel to access it. It's commonly called a power drain and it will not work if the battery is still attached.

It's doubtful that a power drain will work in your case but you never know. It has helped me on other occasions. 

This is what I see when I right-click on Ethernet, so I guess it's enabled since it offers to disable it?

I'm pretty sure this is the correct ethernet, and I downloaded it. Now it's asking if I want to modify, repair, or remove the program for Killer Ethernet Performance Drive Suite UWD. What should I do? @mattyb3 

Unfortunately, I don't have the tools to remove the back panel so I couldn't remove the battery. In any case, it didn't work. I'm going to try the third method now.

I tried the third and fourth methods and I still can't get ethernet to work again.

Bummer. You could try reinstalling Windows again. Maybe something went wrong with the last installation. 
I’d try another Ethernet cable just to rule it out and also reset your modem. Switch it off for 5 mins, unplug it from your laptop and try it again.