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The given expression to expand is:

\[
(-x + 2)(x^2 + 9x - 2)
\]

### Step-by-Step Expansion:

1. Apply the distributive property (FOIL method), where each term in the first binomial \((-x + 2)\) is multiplied by each term in the second binomial \((x^2 + 9x - 2)\).

#### Expanding \( -x \times (x^2 + 9x - 2) \):

\[
- x \times x^2 = -x^3
\]
\[
- x \times 9x = -9x^2
\]
\[
- x \times (-2) = 2x
\]

So, the first part of the expansion is:

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

#### Expanding \( 2 \times (x^2 + 9x - 2) \):

\[
2 \times x^2 = 2x^2
\]
\[
2 \times 9x = 18x
\]
\[
2 \times (-2) = -4
\]

So, the second part of the expansion is:

\[
2x^2 + 18x - 4
\]

### Combining Like Terms:

Now, we combine the like terms from both parts:

\[
- x^3 + (-9x^2 + 2x^2) + (2x + 18x) - 4
\]

This simplifies to:

\[
- x^3 - 7x^2 + 20x - 4
\]

### Final Answer:

The expanded polynomial in standard form is:

\[
-x^3 - 7x^2 + 20x - 4
\]

---

Would you like any further details or explanations on any step?

Here are 5 related questions you might find helpful:
1. How can you verify if the expanded polynomial is correct?
2. What is the difference between standard form and factored form of a polynomial?
3. How would the expansion change if the signs in the original binomials were different?
4. Can you use synthetic division to simplify the expanded form?
5. How does the degree of the polynomial change after expansion?

**Tip:** When expanding binomials, always organize your work carefully and combine like terms systematically to avoid errors!

Expand the expression (-x + 2)(x^2 + 9x - 2) to a polynomial in standard form.

Learn how to expand the polynomial (-x + 2)(x^2 + 9x - 2) step-by-step using the distributive property. This algebraic problem covers key concepts of polynomial multiplication and combining like terms. Suitable for students in Grades 9-10.

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