Final answer:
The passage contains a mythological allusion to Hercules, indicating the extreme difficulty of the task at hand, and the 'Work without Hope' allusion likely refers to Coleridge's poem about laboring without purpose.
Step-by-step explanation:
The type of allusion used in the passage where the student's mother demands that the room be cleaned before going to the dance is a mythological allusion. The phrase 'A Herculean task' refers to Hercules, a character from Greek mythology known for his superhuman strength and the twelve labors he was compelled to perform. Describing the room cleaning as such suggests that it is an extremely difficult or almost impossible task to complete, given the probable state of disarray in the student's room.
As for the allusion from "Work without Hope", it likely refers to a poem by Samuel Taylor Coleridge which addresses the concept of working without joy or the prospects of success. In the context of this poem, the allusion would emphasize the theme of despair and fruitlessness that can come from laboring without a sense of purpose or hope for achievement.