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The problem asks to find \((f * g)(x)\), which refers to the product of the two functions \(f(x)\) and \(g(x)\). Let's break it down step by step:

Given:
\[
f(x) = 3x^2 - 1
\]
\[
g(x) = 3x - 2
\]

The product of \(f(x)\) and \(g(x)\) is calculated by multiplying the two functions:
\[
(f * g)(x) = f(x) \cdot g(x)
\]

Now, multiply the expressions:
\[
(3x^2 - 1)(3x - 2)
\]

Use the distributive property (FOIL method):

1. Multiply \(3x^2 \cdot 3x = 9x^3\)
2. Multiply \(3x^2 \cdot (-2) = -6x^2\)
3. Multiply \((-1) \cdot 3x = -3x\)
4. Multiply \((-1) \cdot (-2) = 2\)

Now, combine these terms:
\[
9x^3 - 6x^2 - 3x + 2
\]

So, the correct expression for \((f * g)(x)\) is:
\[
9x^3 - 6x^2 - 3x + 2
\]

Now, let's match this with the options provided:

**Correct answer: Option b) \(9x^3 - 6x^2 + 3x - 2\)**.

Let me know if you'd like more details or have questions! Here are five related questions:

1. How do you expand binomials using the distributive property?
2. What is the difference between multiplying functions and adding them?
3. Can you explain the FOIL method and how it applies here?
4. What happens if you change \(f(x)\) or \(g(x)\)? How does the product change?
5. Can you factor the expression \(9x^3 - 6x^2 - 3x + 2\)?

**Tip**: Always double-check signs during multiplication to avoid common mistakes!

Escoja la opción correcta: (f * g)(x) es igual a _____ si f(x) = 3x^2 - 1 y g(x) = 3x - 2.

Discover how to find the product of two polynomials f(x) = 3x^2 - 1 and g(x) = 3x - 2 using the distributive property. This problem covers key algebraic concepts and is suitable for students in Grades 9-12.

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