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The given problem asks for a polynomial \( P(x) \) of degree 4 with the following conditions:

- It has a root of multiplicity 2 at \( x = 2 \).
- It has roots of multiplicity 1 at \( x = 0 \) and \( x = -4 \).
- It passes through the point \( (5, 243) \).

### Step-by-Step Solution:

1. **General Form of the Polynomial**:  
   Since the root at \( x = 2 \) has multiplicity 2, the factor associated with it is \( (x - 2)^2 \).  
   The roots at \( x = 0 \) and \( x = -4 \) have multiplicity 1, so their factors are \( x \) and \( (x + 4) \), respectively.

   Therefore, the polynomial can be written in the form:
   \[
   P(x) = a(x - 2)^2 x (x + 4)
   \]
   where \( a \) is a constant to be determined.

2. **Using the Point \( (5, 243) \)**:  
   The polynomial passes through the point \( (5, 243) \), so we can use this information to solve for \( a \).  
   Substituting \( x = 5 \) and \( P(x) = 243 \) into the equation:
   \[
   243 = a(5 - 2)^2 (5)(5 + 4)
   \]
   Simplifying:
   \[
   243 = a(3)^2 (5)(9)
   \]
   \[
   243 = a(9)(5)(9)
   \]
   \[
   243 = a \times 405
   \]
   Solving for \( a \):
   \[
   a = \frac{243}{405} = \frac{9}{15} = \frac{3}{5}
   \]

3. **Final Polynomial**:  
   Substituting \( a = \frac{3}{5} \) back into the general form of the polynomial:
   \[
   P(x) = \frac{3}{5}(x - 2)^2 x (x + 4)
   \]

### Final Answer:
\[
P(x) = \frac{3}{5}(x - 2)^2 x (x + 4)
\]

Let me know if you'd like more details or further clarifications!

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Here are five related questions that expand on this topic:
1. How do you determine the degree of a polynomial from its roots and multiplicities?
2. What is the significance of multiplicity in polynomial roots?
3. How does passing through a specific point help in finding the leading coefficient?
4. How can we check if a polynomial matches certain given conditions?
5. Can the process change if a polynomial is factored over complex numbers?

**Tip**: When working with polynomial roots and multiplicities, always pay attention to how each root affects the shape and behavior of the graph at that point!

The polynomial of degree 4, P(x), has a root of multiplicity 2 at x = 2 and roots of multiplicity 1 at x = 0 and x = -4. It goes through the point (5, 243). Find a formula for P(x).

Learn how to find a degree 4 polynomial with roots of multiplicity 2 at x = 2, and multiplicity 1 at x = 0 and x = -4. Includes step-by-step instructions and explanation, perfect for Grades 10-12 students.

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