To find the zeros of the function [tex]\( g(x) = -3x^4(x+2)^3(x+4)^2 \)[/tex], we need to set [tex]\( g(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex].
### Step-by-step Solution:
1. Identify the factors of the function:
The function [tex]\( g(x) \)[/tex] is given as:
[tex]\[
g(x) = -3x^4(x+2)^3(x+4)^2
\][/tex]
Each factor must be set to zero and solved for [tex]\( x \)[/tex].
2. Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x^4 = 0
\][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[
x = 0
\][/tex]
[tex]\[
(x+2)^3 = 0
\][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[
x = -2
\][/tex]
[tex]\[
(x+4)^2 = 0
\][/tex]
Solving for [tex]\( x \)[/tex] gives:
[tex]\[
x = -4
\][/tex]
3. List the zeros:
The zeros of the function [tex]\( g(x) \)[/tex] are:
[tex]\[
0, -2, -4
\][/tex]
### Answer for Part 1:
The zero(s) of [tex]\( g \)[/tex]:
[tex]\[
0, -2, -4
\][/tex]
