The roots of a polynomial function are the values of x for which the function equals zero. In the given function f(x) = (x - 3)^4 * (x + 6)^2, the roots are the values of x for which either (x - 3) equals zero or (x + 6) equals zero.
Let's start by finding the first root:
Set (x - 3) equal to zero, and solve for x.
So, we get x = 3 is a root.
Then, set (x + 6) equal to zero, and solve for x.
So, we get x = -6 is another root.
The multiplicity of a root is determined by the power of the factor in the polynomial that produces the root. In the given function, the power of (x - 3) is 4 and the power of (x + 6) is 2.
So, the root 3 has a multiplicity of 4, and the root -6 has a multiplicity of 2.
Therefore, the correct option is 3 with multiplicity 4 and -6 with multiplicity 2.
Answer: 3 with multiplicity 2 and –6 with multiplicity 4