A quartic polynomial function with roots $-3$, $-2$, and $4$ (with multiplicity 2 for the root at $4$) can be written as:

$f(x) = a(x + 3)(x + 2)(x - 4)^2$

Here’s the step-by-step explanation:

1. The root $-3$ corresponds to the factor $(x + 3)$.
2. The root $-2$ corresponds to the factor $(x + 2)$.
3. The root $4$ with multiplicity 2 corresponds to the factor $(x - 4)^2$.

If you expand this expression:

$f(x) = a(x + 3)(x + 2)(x - 4)^2$

The constant $a$ is a leading coefficient that can be any nonzero constant. For simplicity, we often set $a = 1$, which gives the polynomial:

$f(x) = (x + 3)(x + 2)(x - 4)^2$

Now, expanding the polynomial:

1. Expand $(x - 4)^2$:

$(x - 4)^2 = x^2 - 8x + 16$

2. Now multiply $(x + 2)$ by $(x^2 - 8x + 16)$:

$(x + 2)(x^2 - 8x + 16) = x^3 - 8x^2 + 16x + 2x^2 - 16x + 32 = x^3 - 6x^2 + 32$

3. Multiply $(x + 3)$ by the result:

$(x + 3)(x^3 - 6x^2 + 32) = x^4 - 6x^3 + 32x + 3x^3 - 18x^2 + 96$

4. Combine like terms:

$f(x) = x^4 - 3x^3 - 18x^2 + 32x + 96$

So, the quartic polynomial is:

$f(x) = x^4 - 3x^3 - 18x^2 + 32x + 96$

Let me know if you'd like further details or have any questions.

Here are 5 related questions:

1. How do you determine the multiplicity of a root?
2. How do leading coefficients affect polynomial graphs?
3. Can a polynomial have more than one root with multiplicity?
4. What is the relationship between the degree of a polynomial and the number of roots?
5. How do you factor polynomials with higher-degree terms?

Tip: When expanding polynomials, handle the terms step by step to avoid mistakes, especially with multiplicity roots.