Final answer:
The expression X^3 - 8y^3 - 6x^2y + 12xy^2 is factored using the difference of cubes for the first two terms and factoring out the common terms for the last two, resulting in (X - 2y)^3.
Step-by-step explanation:
The student is asking to factorize the algebraic expression X^3 - 8y^3 - 6x^2y + 12xy^2. To factorize this expression, we should look for common patterns and use algebraic identities where applicable. Recognizing the first two terms as a difference of cubes, we can write X^3 - 8y^3 as (X - 2y)(X^2 + 2yX + 4y^2). For the remaining terms -6x^2y + 12xy^2, we can factor out a common 6xy to get -6xy(X - 2y). So our factored expression combines these factors:
(X - 2y)(X^2 + 2yX + 4y^2 - 6xy).
Further simplifying the second factor by combining like terms, X^2 + 2yX + 4y^2 - 6xy becomes X^2 - 4yX + 4y^2, which is a perfect square, (X - 2y)^2. Thus, the fully factored expression is (X - 2y)(X - 2y)^2 or (X - 2y)^3.