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To solve the expression \( (4x - 1)(-x^2 + 6x + 9) \) and expand it into a polynomial in standard form, we can apply the distributive property (also known as FOIL in this case) by multiplying each term in the first binomial by each term in the second trinomial.

1. Multiply \( 4x \) by each term in \( (-x^2 + 6x + 9) \):
   \[
   4x \cdot (-x^2) = -4x^3
   \]
   \[
   4x \cdot 6x = 24x^2
   \]
   \[
   4x \cdot 9 = 36x
   \]

2. Multiply \( -1 \) by each term in \( (-x^2 + 6x + 9) \):
   \[
   -1 \cdot (-x^2) = x^2
   \]
   \[
   -1 \cdot 6x = -6x
   \]
   \[
   -1 \cdot 9 = -9
   \]

3. Now, combine all the terms:
   \[
   -4x^3 + 24x^2 + 36x + x^2 - 6x - 9
   \]

4. Combine like terms:
   \[
   -4x^3 + (24x^2 + x^2) + (36x - 6x) - 9
   \]
   \[
   -4x^3 + 25x^2 + 30x - 9
   \]

Thus, the expanded polynomial is:
\[
-4x^3 + 25x^2 + 30x - 9
\]

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

Here are five related questions:
1. How do you apply the distributive property to expressions with more than two terms?
2. What are the common mistakes to avoid when expanding polynomials?
3. How do you identify like terms in a polynomial?
4. How can you factor a polynomial after expanding it?
5. What are the steps to simplify polynomials in higher dimensions?

**Tip:** When expanding expressions, always organize your terms carefully to avoid missing or miscombining like terms!

Expand the expression to a polynomial in standard form: (4x - 1)(-x^2 + 6x + 9)

Learn how to expand the expression (4x - 1)(-x^2 + 6x + 9) into a standard polynomial form with step-by-step instructions. This problem utilizes algebraic concepts including the distributive property and FOIL method. Suitable for students in Grades 7-9.

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