Final answer:
The correct pathways D.trenchi can use for motility are flagellar and ciliary movement, which are essential for the movement of bacteria and eukaryotic cells respectively.
Step-by-step explanation:
The motility pathways that D.trenchi can use for movement are flagellar movement and ciliary movement. Bacterial and eukaryotic cells can exhibit motility through the use of flagella, which function like a propeller to move the cell, and cilia, which are structurally similar to flagella but beat in a coordinated rhythm to propel eukaryotic cells or to move fluids over their surface. The motion of flagella and cilia is powered by the energy from adenosine triphosphate (ATP). While bacteria like E. coli demonstrate peritrichous flagella for movement, eukaryotic cells, such as those of certain protozoans or human sperm cells, use flagella and cilia for their motility.
The correct answer to the question is therefore option (d) Flagellar and ciliary movement.