Final answer:
The given function has a hole at x = 1/2, which does not match any of the provided options. No other factors create holes that are present within the provided options. Therefore, none of the provided options (a, b, c, d) are correct regarding the location of holes in the function.
Step-by-step explanation:
To find the holes of the function ((2x-1)(X+4)²(X-1)(3x+2))/(3x(X-4)(2x-1)³), we need to look for values of x that lead to an undefined expression, typically where the denominator is zero. Holes occur when a factor is in both the numerator and the denominator and can be canceled out, but still, create an undefined point for that specific x value.
While simplifying the function, we notice that the factor (2x-1) appears in both the numerator and in the denominator, specifically it's squared in the numerator as part of ((2x-1)(2x-1)) and cubed in the denominator as (2x-1)³. To find the locations of holes, we set the factor equal to zero and solve for x: 2x - 1 = 0, resulting in x = 1/2. However, since this value does not appear in the given options (a-d), we look for other factors that can be canceled.
Next, we consider the factor 3x in the denominator, which is not canceled by any factor in the numerator. Setting 3x = 0 gives us x = 0, which doesn't create a hole but is a vertical asymptote since the numerator does not become zero at x = 0.
Finally, we can see that no other factors that create holes are present in this function. Therefore, none of the provided options correctly identify a hole of the given function.