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The problem asks for a polynomial of degree 3 that has \( x \)-intercepts at \( x = -4 \), \( x = -3 \), and \( x = 9 \). These intercepts tell us that the polynomial can be factored as:

\[
P(x) = a(x + 4)(x + 3)(x - 9)
\]

Here, \( a \) is a constant that scales the polynomial but does not change the intercepts.

Looking at the answer choices:

1. **\( f(x) = 5(x - 4)(x + 3)(x + 9) \)**: This does not match because the factor \( (x - 4) \) should be \( (x + 4) \), and \( (x + 9) \) should be \( (x - 9) \).
2. **\( g(x) = 2(x - 4)(x - 3)(x + 9) \)**: Same issue with the factors.
3. **\( m(x) = 5(x + 4)(x - 3)(x - 9) \)**: This has the correct factors, except for the second factor, which should be \( (x + 3) \), not \( (x - 3) \).
4. **\( n(x) = 2(x + 4)(x + 3)(x - 9) \)**: This is the correct one! The intercepts match perfectly, and the constant factor 2 does not affect the intercepts.

Thus, the correct polynomial is:

\[
n(x) = 2(x + 4)(x + 3)(x - 9)
\]

Do you want more details on how I verified the factors or have any other questions?

Here are 5 relative questions for further exploration:
1. How do the signs of the factors affect the location of the \(x\)-intercepts?
2. What is the effect of changing the constant factor \(a\) on the graph of the polynomial?
3. How would you determine the degree of a polynomial from its equation?
4. How can you expand the factored form of the polynomial to get its standard form?
5. What is the relationship between the degree of a polynomial and the number of its intercepts?

**Tip**: When working with factored polynomials, each factor represents an \(x\)-intercept or root, and the signs inside the factors are critical to determining the correct intercepts!

A polynomial of degree 3 has x-intercepts at x = -4, x = -3, and x = 9. Which of the following polynomials represents this scenario?

Find out how to determine the polynomial with x-intercepts at x = -4, x = -3, and x = 9. This problem covers factoring and the use of intercepts in polynomials. Suitable for students in Grades 10-12.

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