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The problem is asking for the simplified expression of \((f \cdot g)(x)\) given:

\[
f(x) = x^2 - 2x - 1 \quad \text{and} \quad g(x) = 2x - 1.
\]

**Step 1: Definition of \((f \cdot g)(x)\)**

By definition of multiplication of functions, we have:

\[
(f \cdot g)(x) = f(x) \cdot g(x).
\]

**Step 2: Substitute the given functions**

\[
(f \cdot g)(x) = (x^2 - 2x - 1) \cdot (2x - 1).
\]

**Step 3: Expand the product**

Use the distributive property (FOIL method):

\[
(x^2 - 2x - 1) \cdot (2x - 1) = x^2(2x - 1) - 2x(2x - 1) - 1(2x - 1).
\]

Now, let's expand each term separately:

- \(x^2 \cdot (2x - 1) = 2x^3 - x^2\),
- \(-2x \cdot (2x - 1) = -4x^2 + 2x\),
- \(-1 \cdot (2x - 1) = -2x + 1\).

**Step 4: Combine all the terms**

\[
(f \cdot g)(x) = 2x^3 - x^2 - 4x^2 + 2x - 2x + 1.
\]

Simplify by combining like terms:

\[
(f \cdot g)(x) = 2x^3 - 5x^2 + 1.
\]

**Step 5: Match with the given options**

The simplified expression \(2x^3 - 5x^2 + 1\) matches option **E**.

Would you like additional details or have any other questions?

### Related Questions:
1. How do you find the sum of two functions, \(f(x)\) and \(g(x)\)?
2. What is the difference between multiplying functions and composing functions?
3. Can you find \((f \circ g)(x)\) for the given \(f(x)\) and \(g(x)\)?
4. How would you factor the resulting polynomial \(2x^3 - 5x^2 + 1\)?
5. What happens when the product of functions results in terms of different degrees?

**Tip:** When multiplying two polynomials, always expand carefully and combine like terms to ensure the expression is fully simplified.

Two functions are defined as follows. f(x) = x^2 - 2x - 1 and g(x) = 2x - 1. What is a simplified expression to represent (f · g)(x)?

Explore how to find the simplified expression for the product of two functions, f(x) = x^2 - 2x - 1 and g(x) = 2x - 1. This problem demonstrates polynomial multiplication and simplification, using distributive properties and function operations. Suitable for students in Grades 9-11.

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