To solve the problem using the Factor Theorem, we check whether $x - 3$ is a factor of $P(x) = -x^4 + 2x^3 + 5x^2 - 18$ by substituting $x = 3$ into the polynomial. If $P(3) = 0$, then $x - 3$ is a factor.

Step-by-step solution:

1. Substitute $x = 3$ into $P(x)$: $P(3) = -(3)^4 + 2(3)^3 + 5(3)^2 - 18$

2. Simplify each term:

   • $-(3)^4 = -81$
   • $2(3)^3 = 54$
   • $5(3)^2 = 45$
   • $-18$ remains as it is.

3. Combine all terms: $P(3) = -81 + 54 + 45 - 18$

4. Perform the addition:

   • $-81 + 54 = -27$
   • $-27 + 45 = 18$
   • $18 - 18 = 0$

5. Since $P(3) = 0$, the Factor Theorem confirms that $x - 3$ is a factor of $P(x)$.

Final Answer:

$x - 3$ is a factor of $P(x)$.

Do you have any questions or want additional details? Here are related questions to consider:

1. How does the Factor Theorem work for higher-degree polynomials?
2. Can $x - c$ be a factor if $P(c) \neq 0$?
3. What are other methods to determine polynomial factors besides substitution?
4. How can synthetic division confirm that $x - 3$ is a factor?
5. How do multiple roots affect the factorization of polynomials?

Tip: Always double-check substitution calculations carefully to avoid small errors that could affect the conclusion.