Final answer:
The viral exit method that does not immediately destroy the host bacterium is referred to as extrusion, also known as budding. This method allows the release of virions while keeping the host cell intact. In the case of bacteriophages, this is part of the lysogenic cycle where the virus genome is integrated into the host's genome.
Step-by-step explanation:
An exit method used by viruses that do not immediately destroy the host bacterium is known as extrusion more commonly referred to as budding. This process is observed in certain animal viruses where virions exit the host cell one by one by capturing a piece of the host's plasma membrane. Unlike the lytic cycle where viruses cause the cell to lyse, or break apart, resulting in cell death, budding allows the cell to remain intact for some time, although it may be functionally compromised. In the context of bacteriophages, a phage that integrates its genome into the host genome without causing immediate cell death undergoes what is known as the lysogenic cycle. During lysogeny, the virus or phage genome becomes a prophage and can remain within the bacterial cell without disrupting its normal activities until it excises under certain environmental conditions and enters the lytic cycle. Therefore, the exit method that doesn't immediately destroy the host bacterium is extrusion or budding, and not lysis, inversion, or excising. These methods either imply the destruction of the host cell or are related to other functions within the viral lifecycle.