To simplify the expression \((4x - 2)(x^2 - 5x + 4)\), we need to distribute (also known as using the distributive property or FOIL method for binomials) each term in the first polynomial \((4x - 2)\) across each term in the second polynomial \((x^2 - 5x + 4)\).

### Step-by-Step Solution

1. **Distribute \(4x\) to each term in \((x^2 - 5x + 4)\):**

\[
4x \cdot x^2 = 4x^3
\]
\[
4x \cdot (-5x) = -20x^2
\]
\[
4x \cdot 4 = 16x
\]

2. **Distribute \(-2\) to each term in \((x^2 - 5x + 4)\):**

\[
-2 \cdot x^2 = -2x^2
\]
\[
-2 \cdot (-5x) = 10x
\]
\[
-2 \cdot 4 = -8
\]

3. **Combine all these terms:**

\[
4x^3 - 20x^2 + 16x - 2x^2 + 10x - 8
\]

4. **Combine like terms:**

   - Combine the \(x^2\) terms: \(-20x^2 - 2x^2 = -22x^2\)
   - Combine the \(x\) terms: \(16x + 10x = 26x\)

5. **Write the polynomial in standard form:**

\[
4x^3 - 22x^2 + 26x - 8
\]

### Final Answer

The simplified expression is:

\[
4x^3 - 22x^2 + 26x - 8
\]

Would you like further details or have any questions about this solution?

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Here are 5 related questions to explore further:

1. What are the steps for multiplying two polynomials?
2. How do you apply the distributive property when multiplying polynomials?
3. How can you use the FOIL method for multiplying binomials?
4. What is the standard form of a polynomial, and why is it useful?
5. How do you determine the degree of a polynomial after multiplication?

**Tip:** When multiplying polynomials, carefully distribute each term to avoid mistakes, and always combine like terms to simplify the final expression.

Learn how to simplify the expression (4x - 2)(x^2 - 5x + 4) step-by-step. This problem involves multiplying polynomials using the distributive property and combining like terms. Suitable for high school students.

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