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We are tasked with finding a possible formula for the polynomial \( P(x) \), which is of degree 5, has leading coefficient 1, and has roots of specified multiplicities.

### Key information:
- **Root of multiplicity 2** at \( x = 4 \).
- **Root of multiplicity 2** at \( x = 0 \).
- **Root of multiplicity 1** at \( x = -1 \).
- The **leading coefficient is 1**.

### Polynomial structure:

If a root has multiplicity \( m \), the factor corresponding to that root will be of the form \( (x - r)^m \). For this polynomial:

- The factor for the root \( x = 4 \) with multiplicity 2 is \( (x - 4)^2 \).
- The factor for the root \( x = 0 \) with multiplicity 2 is \( x^2 \) (since \( x - 0 = x \)).
- The factor for the root \( x = -1 \) with multiplicity 1 is \( (x + 1) \).

### Assembling the polynomial:
Since the leading coefficient is 1, the polynomial can be written as:

\[
P(x) = (x - 4)^2 \cdot x^2 \cdot (x + 1)
\]

This is a possible formula for the polynomial \( P(x) \).

Would you like me to expand this polynomial or provide further details?

### Related questions:
1. How do you expand the polynomial \( (x - 4)^2 \cdot x^2 \cdot (x + 1) \)?
2. What does the term "multiplicity" of a root mean in polynomial functions?
3. How do the multiplicities affect the shape of the graph of the polynomial?
4. Can a polynomial have a root of higher multiplicity than its degree?
5. How does the leading coefficient affect the end behavior of a polynomial?

**Tip:** The sum of the multiplicities of the roots of a polynomial equals the degree of the polynomial.

The polynomial of degree 5, P(x), has leading coefficient 1, has roots of multiplicity 2 at x = 4 and x = 0, and a root of multiplicity 1 at x = -1. Find a possible formula for P(x).

Find the possible formula for a polynomial P(x) of degree 5 with a leading coefficient of 1, roots at x = 4, x = 0, and x = -1. This example involves understanding the concept of roots and their multiplicities. Suitable for students in Grades 9-12.

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