Final answer:
The binomial that completes the factorization of the expression 24t³u - 81u is (2t - 3), resulting in the factorization 3u(2t - 3)(4t² + 6t + 9).
Step-by-step explanation:
The student is asked to find the binomial that completes the factorization of a given expression. The expression to be factored is 24t³u - 81u. This expression can be factored by taking out the common factor 3u, which results in 3u(8t³ - 27). The term 8t³ - 27 is a difference of two cubes, which can be factored further into (2t - 3)(4t² + 6t + 9). Therefore, the missing binomial that the student is looking to find is (2t - 3), completing the factorization as 3u(2t - 3)(4t² + 6t + 9).