Question

Determine the multiplicity of the roots of the function k(x)=x(x+2)cubed(x+4)squared(x-5)cubed

Answer 1

The degree of the (x - h), whether 1, 2, or 3, says how many multiplicities that linear factor has.

So,
0
-2 (m of 3)
-4 (m of 2)
5 (m of 3)

Answer 2

Multiplicity is the power on the factor.
x(x + 2)^3(X + 4)^2(x - 5)^3

x = 0 with multiplicity 1
x = -2 with multiplicity 3
x = -4 with multiplicity 2
x = 5 with multiplicity 3