Final answer:
Factoring the polynomial involves grouping and factoring out common factors from those groups. The factored form of the polynomial is (-4y + 3x)(y + 3) which does not match the given options exactly, indicating the provided options might be incorrect or there is a typo in the question.
Step-by-step explanation:
To factor the polynomial -4y² + 3xy - 12y + 9x, we can start by grouping terms that have common factors. By inspecting the terms, we see that the first two terms have a common factor of y, and the last two terms have a common factor of 3. Thus, we can arrange and group the terms like this:
(-4y² - 12y) + (3xy + 9x)
Now, we can factor a -4y out of the first group and a 3x out of the second group:
-4y(y + 3) + 3x(y + 3)
Now, we see that the (y + 3) is a common factor of both terms. So the final factored form of the polynomial is:
-4y + 3x(y + 3)
This does not match any of the given multiple-choice options exactly. Therefore, the given answer choices might be incorrect, or there may have been a typo in the initial question. If the student insists on an answer from the provided options, then it seems there is a need for clarification.