Final answer:
The function f(x) = (x - 6)^2(x + 2)^2 has four roots in total, with two repeated roots at x = 6 and two repeated roots at x = -2.
Step-by-step explanation:
The function given is f(x) = (x - 6)^2(x + 2)^2. To find how many roots there are, we need to look at the factors of the function. Both (x - 6) and (x + 2) are squared, which means each of them generates a root at their respective x-values where the factor equals zero.
For the factor (x - 6)^2, the root is x = 6. Since it is squared, this is a repeated root, meaning it counts twice. Similarly, the factor (x + 2)^2 gives us a root at x = -2, which is also a repeated root.
Therefore, the function has a total of four roots: two at x = 6 and two at x = -2.