Final answer:
To complete the expression X^2-12x+ ___, we need to determine the constant term that allows factoring into (x-5)(x-7). By analyzing the given binomial (x-5), we conclude that to achieve the -12x term upon expansion, the missing factors are 5 and 7, thus the constant term is 35.
Step-by-step explanation:
To fill in the blanks for the expression X^2-12x+ ___ so that it can be factored into (x-5)(x+__), we must first understand that this is a problem related to completing the square and factoring quadratic expressions.
We are given one binomial, (x-5), which means the other must be (x-7) to get the middle term -12x when expanded (-5x - 7x = -12x). The constant term of the first binomial (-5) multiplied by that of the second binomial (-7) should give us the constant term we need to complete the square, which is 35. Therefore, the completed expression becomes X^2 - 12x + 35, and when factored, it is indeed (x-5)(x-7).