To solve the problem, let's follow these steps:
1. Define the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = x(x-2)(x+6)^2 \][/tex]
2. Substitute [tex]\( x \)[/tex] with [tex]\( 7 - w \)[/tex] in the function:
[tex]\[ g(7 - w) = (7 - w)((7 - w) - 2)((7 - w) + 6)^2 \][/tex]
3. Simplify the expression:
Substitute the values step-by-step:
[tex]\[ g(7 - w) = (7 - w)((7 - w) - 2)((7 - w) + 6)^2 \][/tex]
[tex]\[ g(7 - w) = (7 - w)(7 - w - 2)((7 - w) + 6)^2 \][/tex]
[tex]\[ g(7 - w) = (7 - w)(5 - w)((13 - w))^2 \][/tex]
4. Set the equation to zero because [tex]\( g(7 - w) = 0 \)[/tex]:
[tex]\[ (7 - w)(5 - w)(13 - w)^2 = 0 \][/tex]
This equation will be zero if any of its factors is zero:
[tex]\[ 7 - w = 0 \quad \text{or} \quad 5 - w = 0 \quad \text{or} \quad (13 - w)^2 = 0 \][/tex]
5. Solve each factor for [tex]\( w \)[/tex]:
[tex]\[ 7 - w = 0 \implies w = 7 \][/tex]
[tex]\[ 5 - w = 0 \implies w = 5 \][/tex]
[tex]\[ (13 - w)^2 = 0 \implies 13 - w = 0 \implies w = 13 \][/tex]
Therefore, the possible values of [tex]\( w \)[/tex] are [tex]\( 7 \)[/tex], [tex]\( 5 \)[/tex], and [tex]\( 13 \)[/tex].
6. Find the sum of all possible values of [tex]\( w \)[/tex]:
[tex]\[ 5 + 7 + 13 = 25 \][/tex]
So, the sum of all possible values of [tex]\( w \)[/tex] is [tex]\( \boxed{25} \)[/tex].
