Question

QM Q2.) Find the multiplicities of the greatest zero, second greatest zero, and the smallest zero.

Answer 1

Method
First check out the zeros (first box). 

Step One.
Put brackets around terms 1 and 2 and another set around terms 3 and 4.

f(x) = (x^3 + 8x^2) - (4x + 32) Watch out that you change the sign on the 32. Minus signs can be devastating.   The idea is that when the brackets are removed, the question will be exactly as it was before you put the brackets there.

Step Two
Pull out any common factors.
f(x) = x^2(x + 8) - 4(x + 8)

Step Three.
If there is a common binomial pull it out as well. This is the hardest part of the problem. The binomials have to be exactly alike (in this case they are).

f(x) = (x + 8) (x^2 - 4) 

Step Four
Factor (x^2 - 4). This is an exact difference of squares. It factors into
x^2 - 4 = (x + 2)(x - 2)

Step five
Find the zeros.
f(x) = (x + 8)(x - 2)(x + 2)

x + 8 = 0
x = - 8

x + 2 = 0
x = - 2

x - 2 = 0
x = 2

I know you didn't ask for all of this, but when someone asks a question, it's always a good idea to check the work done. In your case it was all correct. That's a very good thing.

Definition
Multiplicity is the number of times a root appears in the solution of a polynomial. 
For example, g(x) = (x -  5)^2 (x - 4)
The five would have a multiplicity of 2 because the equation is really
g(x) = (x - 5)(x - 5) ( x - 4). There are 2 roots that are the same (both are 5).
4 has a multiplicity of 1 in this example.

Answer
The largest root is 2. It has a multiplicity of 1
The second largest root is -2. It has a multiplicity of 1
The tiniest root is - 8 . It has a multiplicity of 1