I'd actually push back a bit on the idea that you're automatically out of luck here. A lot of people don't realize that warranty periods and consumer protection laws are two different things. Most states have laws about how long a product is expected to last based on what you paid for it - an $800 washing machine failing one month after a one-year warranty is genuinely suspicious and might actually be covered under what's called implied warranty of merchantability. The timeframe varies by state, but many places give you protection for several years after purchase, not just the manufacturer's warranty period. It's worth digging into your state's consumer protection laws before you accept that $400 repair bill.
I had something similar happen with a refrigerator a while back. It quit working about six weeks after the warranty expired, and the repair place quoted me almost half the original price. I looked into my state's laws and found out I had about four years of protection from the date of purchase. I contacted the manufacturer with that info and photos of the purchase date, and honestly they settled pretty quickly - offered to cover most of the repair. They're banking on people not knowing their rights, so they just say no at first.
Document everything: take photos of the purchase receipt with the date visible, the warranty card, and the repair quote. Send a formal letter (email works) to Maytag explaining that the machine failed shortly after warranty expiration and you believe this qualifies as a defect under implied warranty laws in your state. Be specific about your state. If they don't respond, small claims court is an option in most places, and companies usually don't want to show up for those. You might not win the full amount, but you've got a real case here that's worth pushing.