Which method of reproduction involves a new cell forming by growing out from a part of the parent cell?

The method of reproduction described involves a new cell forming by growing out from a part of the parent cell, which is characteristic of budding. In this process, a small outgrowth, or bud, develops on the surface of the parent organism, and as it grows, it eventually separates and becomes a new individual. This type of reproduction is commonly seen in certain organisms such as yeast and hydra.

Fission, fragmentation, and binary fission refer to different reproductive strategies. Fission typically involves the splitting of the parent into two or more parts, while fragmentation is the breaking of an organism into pieces, which later regenerate into new individuals. Binary fission is a specific form of fission where a single parent cell divides into two equal parts, commonly seen in bacteria. Therefore, the details of this question align clearly with the process of budding and emphasize how new organisms can arise from the growth of the parent organism rather than through splitting or fragmentation.