Given the conditions for the polynomial $P(x)$:

• The polynomial is of degree 8.
• The leading coefficient is 1.
• There is a root of multiplicity 1 at $x = 0$.
• There is a root of multiplicity 3 at $x = -4$.
• There is a root of multiplicity 4 at $x = -2$.

We can express $P(x)$ in its factored form as follows:

$P(x) = a(x - 0)^1(x + 4)^3(x + 2)^4$

Given that the leading coefficient is 1, the constant $a = 1$. Thus, the polynomial simplifies to:

$P(x) = x(x + 4)^3(x + 2)^4$

This is the polynomial $P(x)$ that satisfies all the given conditions.

Would you like more details or have any questions?

Here are some related questions:

1. How can the degree of a polynomial be determined by its roots?
2. What is the significance of the leading coefficient in a polynomial?
3. How do you find the multiplicity of a root in a polynomial?
4. How would the polynomial change if the leading coefficient was different?
5. Can you graph the polynomial to verify its roots and multiplicities?

Tip: Always check that the sum of the multiplicities of the roots matches the degree of the polynomial to ensure accuracy.