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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.