If the deal has been approved by a lender and you have signed paperwork and then driven the veicle off the lot they can not unwind the deal. Their mistake they have to eat it if you are delivered in the vehicle. Unless you signed something giving them recourse (usually dealers have forms for salvage title vehicles that allow them to unwind the deal if they discover salvage/branded title)If they just misbooked the trade and made the mistake it's their problem. As long as you didn't misrepresent the trade you should be fine.
Sure the dealer can ask for more money and you can refuse. Tell them a deal is a deal, and you expect them to honor the contract they signed. You payed them and hopefully you are in possession of the car. If so, I see nothing they can do, until you come back for service. Then they may try to recoup their loss. So be aware of that. If they made an honest mistake and it is very little money I would make it right if it were me. But legally I think you do not have to. Your decision.
If you signed it you are bound to it, even if you didn't read it beforehand. If you haven't signed it then they are responsible for fixing the mistake before you sign it if they still want your business. If you signed it and later came to a problem where you realized the contract was ambiguous it will work to your advantage under the rule of contra proferentum.
The dealer will pay off the loan but it t will cost you a fortune by a low tradein price and the dealer refinancing the original loan payoff. Don't do it. You can come out much better by finding a buyer yourself and paying off the loan with some of the proceeds. You are in a much better position to make a better deal or get a discount on the next car you want with more cash in your pocket. This can be a big bucks difference depending on the value of your car and the amount you owe.
The cast of Play Your Cards Right - 1980 includes: Sophie Allisstone as Dolly Dealer Lesley Anderson as Dolly Dealer Alison Bell as Dolly Dealer Camilla Blair as Dolly Dealer Annalise Braakensiek as Dolly Dealer Vicki Brattle as Dolly Dealer Carly Carter as Dolly Dealer Zena Clifton as Dolly Dealer Dickie Davies as himself Gillian Duxbury as Dolly Dealer Gillian Elvins as Dolly Dealer Debbie Flett as Dolly Dealer Bruce Forsyth as Himself - Host Brenda Haldane as Dolly Dealer Denise Kemp as Dolly Dealer Charlie Maloney as Dolly Dealer Wendy Marler as Dolly Dealer Robin Martyne as Dolly Dealer John Melainey as Male Model Jan Michelle as Dolly Dealer Claire Rayner as herself Maxine Restall as Dolly Dealer Yvonne Younger as Dolly Dealer
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