Let's simplify the given expression step-by-step:

The expression is:

\[
(4x^4 - 3x^2 + 6x - 9)(-x)^2 + 4x - 2
\]

### Step 1: Simplify \((-x)^2\)

\[
(-x)^2 = x^2
\]

### Step 2: Substitute \(x^2\) into the expression

\[
(4x^4 - 3x^2 + 6x - 9) \cdot x^2 + 4x - 2
\]

### Step 3: Distribute \(x^2\) to each term inside the parentheses

\[
(4x^4) \cdot x^2 - (3x^2) \cdot x^2 + (6x) \cdot x^2 - (9) \cdot x^2 + 4x - 2
\]

Simplifying each term:

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

### Step 4: Combine all the terms

\[
4x^6 - 3x^4 + 6x^3 - 9x^2 + 4x - 2
\]

This is the simplified form of the given expression:

\[
\boxed{4x^6 - 3x^4 + 6x^3 - 9x^2 + 4x - 2}
\]

Would you like more details or have any questions?

Here are some related questions you might consider:

1. How do you factor higher-degree polynomials?
2. What are common techniques for simplifying expressions involving powers?
3. How do you determine the degree of a polynomial?
4. What are the key differences between expanding and factoring expressions?
5. How do you find the roots of a polynomial equation?

**Tip:** Always distribute terms carefully, especially when dealing with negative signs and exponents!

Learn how to simplify the expression (4x^4 - 3x^2 + 6x - 9)(-x)^2 + 4x - 2 step-by-step with a detailed solution. This problem involves polynomials, exponentiation, and distribution. Suitable for high school students.

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