Here are different ways to use the word 많다.
많은 = adjective (attributive, placed before a noun or pronoun). 많은 일을 했다 = did a lot of work.
많이 = adverb. 일을 많이 했다 = worked a lot (structurally "did work a lot", but English doesn't work well this way).
The others are sentence ending forms which can only come at the end of a sentence as a predicative adjective.
많습니다 (polite and formal), 많다 (impersonal), 많아요 (polite), 많아 (casual).
The predicative adjectives function like verbs. Think of them as verbs like "teem" or "proliferate" in English.
For example, here are more ways to say "There are a lot of people in the park." in Korean.
공원에 많은 사람이 있습니다 / 많다 / 많아요 / 많아.. (using the adjective - "a lot of" people)
공원에 사람이 많이 있습니다. (using the adverb - structurally similar to "there are people a lot")
공원에 사람이 많습니다. (using the sentence ending form - structurally similar to "teems (많다) with people")
You can construct similar structures for your example too.
방과후에 (or 학교 끝난 후에) 저는 많은 숙제를 해애 돼요.
방과후에 저는 숙제를 많이 해야 돼요.
방과후에 저는 해야 할 숙제가 많아요.