you can almost definitely get your car looking like new again (assuming that scratches from the car wash are the only problem), but don't expect it to happen in 15 minutes with
scratchx.
Having no idea how bad the scratches are, no one can tell you how easy it's going to be or what product will probably work the best.
Give the scratchx a try as it's a good hand application product as TOGWT said. If you're not happy with how it looks, you have two choices. Try again (sometimes a second attempt with the same product is all you need), or come back here and spend a little time researching about polishing out swirls and scratches.
Based on your description, you can probably fix the damage just fine, but have patience. There's a technique that goes along with everything you read here, and you're not going to get 100% perfect results the first time (unless you're way better at this than me

).
John