To solve this problem, we need to understand the gene regulation mechanism of the lactose operon in $E.$coli, specifically involving the $$lac I$$ gene product.$\newline \bullet$The lactose operon is a classic example of an inducible operon, which is typically off but can be turned on in the presence of an inducer.$\newline \bullet$The $$lac I$$ gene encodes the repressor protein, which binds to the operator region of the operon and prevents transcription.$\newline \bullet$In the absence of lactose, the repressor binds to the operator, blocking RNA polymerase from transcribing the structural genes.$\newline \bullet$When lactose is present, it acts as an inducer by binding to the repressor protein, causing a conformational change that prevents the repressor from binding to the operator.$\newline \bullet$This allows RNA polymerase to access the promoter and transcribe the genes necessary for lactose metabolism.$\newline$Based on this understanding, the regulation is:$\newline \bullet$Negative and inducible, because the repressor protein prevents transcription in the absence of lactose, and the presence of lactose induces transcription by inactivating the repressor.$\newline$Therefore, the correct option is Option 2: Negative and inducible because repressor protein prevents transcription.