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A deed in lieu is basically deeding the property back to the mortgage lender in lieu of the lender filing foreclosure to get the property back when the borrower is in default.
A short sale is selling the property for less than the mortgage amount.
Ex: the mortgage balance is $200K but the market value is only $175K. You must obtain approval from the lender to this type of sale.
The balance due may be waived, or may be settled in some way depending on what the lender requires.