To solve the polynomial equation [tex]\( f(x) = -2x^4 + 18x^3 - 54x^2 + 162x - 324 \)[/tex], we need to find the values of [tex]\( x \)[/tex] that satisfy this equation.
### Step-by-Step Solution:
1. Identify the polynomial equation:
[tex]\[
f(x) = -2x^4 + 18x^3 - 54x^2 + 162x - 324
\][/tex]
2. Equating the polynomial to zero:
[tex]\[
-2x^4 + 18x^3 - 54x^2 + 162x - 324 = 0
\][/tex]
3. Find the roots:
We need to find the solutions for [tex]\( x \)[/tex] that satisfy this polynomial equation.
Without detailed algebraic manipulations, we can list out the roots directly based on the problem's parameter and the context given.
4. Roots of the polynomial:
Upon solving the polynomial equation, the roots are:
[tex]\[
x = 3, 6, 3i, -3i
\][/tex]
### Verification with the provided options:
- Option A: [tex]\( x = \{-3, 6, 3i, -3i\} \)[/tex]
- Option B: [tex]\( x = \{3, 6, 3i, -3i\} \)[/tex]
- Option C: [tex]\( x = \{-2, 3, 6, 3i, -3i\} \)[/tex]
- Option D: [tex]\( x = \{-6, -3, 3i, -3i\} \)[/tex]
From the roots we have found [tex]\( x = 3, 6, 3i, -3i \)[/tex], which correspond to "Option B".
Therefore, the correct option is:
[tex]\[
\boxed{B}
\][/tex]
