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The function given is:

\[
f(x) = (3x^2 + 3x - 6)(x^2 + 4)
\]

To find the multiplicities of the roots (zeros) of this function, we need to solve for \(x\) by setting \(f(x) = 0\) and analyzing the factors.

### Step 1: Find the roots of each factor.

- **First factor: \(3x^2 + 3x - 6 = 0\)**

  We solve the quadratic equation \(3x^2 + 3x - 6 = 0\) by using the quadratic formula:
  
  \[
  x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  \]
  where \(a = 3\), \(b = 3\), and \(c = -6\).

  \[
  x = \frac{-3 \pm \sqrt{3^2 - 4(3)(-6)}}{2(3)} = \frac{-3 \pm \sqrt{9 + 72}}{6} = \frac{-3 \pm \sqrt{81}}{6}
  \]

  \[
  x = \frac{-3 \pm 9}{6}
  \]

  Thus, the two solutions are:

  \[
  x_1 = \frac{-3 + 9}{6} = 1, \quad x_2 = \frac{-3 - 9}{6} = -2
  \]

- **Second factor: \(x^2 + 4 = 0\)**

  Solving \(x^2 + 4 = 0\):

  \[
  x^2 = -4 \quad \Rightarrow \quad x = \pm 2i
  \]

  These are complex roots: \(x = 2i\) and \(x = -2i\).

### Step 2: Determine the multiplicities.

- The factor \(3x^2 + 3x - 6 = 0\) is a quadratic equation, meaning both real roots \(x = 1\) and \(x = -2\) have multiplicity 1.
- The factor \(x^2 + 4 = 0\) is also quadratic, meaning both complex roots \(x = 2i\) and \(x = -2i\) have multiplicity 1.

### Summary of Roots and Their Multiplicities:

- Real root \(x = 1\), multiplicity 1.
- Real root \(x = -2\), multiplicity 1.
- Complex root \(x = 2i\), multiplicity 1.
- Complex root \(x = -2i\), multiplicity 1.

Would you like further explanation or details on any step?

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Here are 5 follow-up questions to deepen your understanding:

1. How would you graph the function \(f(x)\) and represent its real and complex roots?
2. How does the nature of complex roots (e.g., \(2i\) and \(-2i\)) affect the graph of the function?
3. What changes in the multiplicity of a root affect the graph at that root's location?
4. How would the solutions change if the coefficient of \(x^2 + 4\) was different (e.g., \(x^2 + c\))?
5. What happens to the roots and multiplicities if we modify the first factor \(3x^2 + 3x - 6\)?

**Tip:** To find the nature of roots in a quadratic equation, the discriminant (\(b^2 - 4ac\)) can reveal if the roots are real or complex.

Find multiplicities of the roots for f(x) = (3x^2 + 3x - 6)(x^2 + 4).

Learn how to find the multiplicities of the roots for the function f(x) = (3x^2 + 3x - 6)(x^2 + 4). This problem involves solving quadratic equations and working with complex numbers. Ideal for students in Grades 10-12.

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