Final answer:
The question seems to contain a typo, but it seeks the missing coefficient of xy in a polynomial resembling a difference of cubes. The proper factorization indicates a coefficient of 21 for xy, which does not match any of the provided options.
Step-by-step explanation:
The student's question pertains to finding the unknown coefficients of terms in a polynomial that is factorable using the binomial theorem. We are given 27x³ - __x²y + xy² – 343y³. Both 27 and -343 are perfect cubes, such that 27 = 3³ and 343 = 7³. This hints that the polynomial is a difference of cubes which can be factored as (a³ - b³) = (a - b)(a² + ab + b²). Plugging in our terms, we get (3x - 7y)(9x² + 21x²y + 49y²). Thus, the coefficient of xy in the polynomial would be 21.
However, 21 is not one of the options provided, and the question likely contains a typo. Unfortunately, given the options within the question and the information on hand, we cannot determine which correct coefficient corresponds to these options. Thus, we cannot confidently provide an answer from the options listed.