Gene regulation governing lactose operon of E.coli that involves the lac I gene product is

Positive and inducible because it can be induced by lactose

Negative and inducible because repressor protein prevents transcription

Negative and repressible because repressor protein prevents transcription.

Feedback inhibition because excess of $straight beta$ -galactosidase can switch off trascription.

Solution

Negative and inducible because repressor protein prevents transcription

Lac I gene produces an inhibitor or repressor and negative regulation of lac operon is induced. The repressor binds to the operator gene and stops its working. Repressor is meant to block the operator gene so that structural genes are unable to form mRNA thus stopping the transcription of genes.