Your approach is correct. But what isn’t shown in the figure explicitly, are the non-inverting and the inverting input terminals.
This case, you can safely assume it’s a negative feedback. Hence, the feedback resistor Rf, is connected to the inverting input terminal.
Therefore, the differential input voltage V+ - V- = -v
Hence, after applying the loop equations as you have done, and assuming the output resistance getting getting swamped by negative feedback, you get the same expression as given in the book.