Wider die laienhafte Deutung von #6:
entertain (v.)late 15c., "to keep up, maintain, to keep (someone) in a certain frame of mind," from Middle French entretenir, from Old French entretenir "hold together, stick together, support" (12c.), from entre- "among" (from Latin inter; see inter-) + tenir "to hold" (from Latin tenere, from PIE root *ten- "to stretch").
Sense of "have a guest" is late 15c.; that of "gratify, amuse" is 1620s. Meaning "to allow (something) to consideration, take into the mind" (of opinions, notions, etc.) is 1610s.
-ment suffix forming nouns, originally from French and representing Latin -mentum, which was added to verb stems sometimes to represent the result or product of the action. French inserts an -e- between the verbal root and the suffix (as in commenc-e-ment from commenc-er; with verbs in ir, -i- is inserted instead (as in sent-i-ment from sentir). Used with English verb stems from 16c. (for example amazement, merriment, which also illustrates the habit of turning -y to -i- before this suffix).