Define the term: chemotaxis literally means moving along a chemical gradient, so the defining feature must involve directed travel.
Reject the distractors: increased random movement is the opposite of directed, so it fails. Increased adhesiveness to the vessel intima describes margination and adhesion, an earlier and separate step in the leukocyte cascade. Increased phagocytosis happens after the neutrophil arrives, so it is downstream, not the essence of chemotaxis.
The answer: unidirectional locomotion of neutrophils toward rising concentrations of chemoattractants such as C5a, LTB4, and chemokines, plus bacterial peptides from outside the body.
Remember the sequence: margination, rolling, adhesion, transmigration, then chemotaxis, then phagocytosis. Chemotaxis is the directed crawl, step by step toward the target.
Reference: inflammation, leukocyte recruitment.