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!

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!