Final answer:
The process involved in the production of a new plant from the clipped stem placed in water is mitosis, as cell division occurs in the apical meristems which are responsible for the growth and development of new roots and shoots in plants.
Step-by-step explanation:
The processes of cell division involved in the production of a new plant from the original plant through vegetative propagation, as described in Patrick's experiment with the spotted house plant, are related to mitosis only. In the process of stem clipping and rooting, specific cells known as parenchyma cells dedifferentiate and are stimulated to divide to organize and form a new apical meristem. Cell division in plants occurs primarily at specific locations called meristems, which are the regions where mitotically active cells produce all the specialized tissues of the organism. Particularly, new roots on the plant cutting grow from root apical meristems, which are areas of intense cell division activity at the root tips. As a result, mitosis is the key process that facilitates the production of new cells that eventually differentiate and develop into roots. These new cells then continue to show vacuolar expansion, leading to the growth of the plant once it is transplanted into soil.