Answer: To determine the total number of roots of the polynomial function f(x) = (x - 6)^2(x + 2)^2, we can look at the powers of the factors in the factored form.
The power of a factor in the factored form represents the multiplicity of the corresponding root. The multiplicity of a root indicates how many times the root appears in the factored form.
In this case, the factor (x - 6) has a power of 2, and the factor (x + 2) also has a power of 2. This means that both the roots x = 6 and x = -2 have a multiplicity of 2.
The total number of roots of a polynomial function is equal to the sum of the multiplicities of all its distinct roots.
In this polynomial function, the root x = 6 has a multiplicity of 2, and the root x = -2 also has a multiplicity of 2. Therefore, the total number of roots is 2 + 2 = 4.
So, the correct answer is that the polynomial function f(x) = (x - 6)^2(x + 2)^2 has a total of 4 roots.