The process you are describing is characteristic of the lytic cycle of viral replication. In the lytic cycle, a virus infects a host cell, replicates its genetic material, assembles new viral particles, and then causes the host cell to burst (lyse), releasing the newly formed viruses to go on and infect other cells.
In contrast, the lysogenic cycle involves the integration of the viral DNA into the host cell's genome, where it can remain dormant for an extended period. During the lysogenic phase, the viral DNA is replicated along with the host DNA during cell division without causing immediate damage to the host. Later, the virus can enter the lytic cycle, leading to cell lysis and the release of new virions.
To summarize, the bursting of the host cell releasing new viruses is a feature of the lytic cycle.