What is the best way to remove this paint scratch

Those scratches look kinda deep, and even lost of paint. Try Meguair's Scratch X it would remove the shallow scratches but where there is lost of paint , it wont fix.

 

I'd suspect it'll be a repaint of the panel. Talk to an auto spray painter - it is possible that they might be able to "improve" it with a "cut & polish".

 

I’d recommend a competent body shop. Just let them work their magic.

 

I got a couple deep scratches out of my paint with a 3m kit from walmart like this one: 3M 39071 Automotive Scratch Removal System. Stupid people or carts!!! ARGH!

It takes about an hour and follow the directions. Maybe watch a few youtube videos too! It will look way worse for a bit and once you get finished it removes WAY more than you would expect it to remove. One was down to the primer on my car but none the less, the other annoying scratches were gone. I think it costs like $15 and would highly recommend the drill kit if you own a drill gun. Good Luck, I was plenty impressed with the results myself. Way better than Living with it! The best thing is that the scratches don't reappear, at least with usual bi-annual waxings. My first...repair happened about 3 years ago and still has not reappeared.