https://file.aiquickdraw.com/mathbot/file/1726838996739tvzk2etw.jpg

The expression in the image is \( (-2xy - 2)^3 \).

Let's expand this expression using the binomial theorem.

We will use the formula for binomial expansion:

\[
(a + b)^n = a^n + n a^{n-1} b + \frac{n(n-1)}{2!} a^{n-2} b^2 + \dots + b^n
\]

For \( (-2xy - 2)^3 \), we have \( a = -2xy \) and \( b = -2 \), with \( n = 3 \). 

Let's expand:

\[
(-2xy - 2)^3 = (-2xy)^3 + 3(-2xy)^2(-2) + 3(-2xy)(-2)^2 + (-2)^3
\]

Now, calculate each term:

1. \( (-2xy)^3 = -8x^3y^3 \)
2. \( 3(-2xy)^2(-2) = 3(4x^2y^2)(-2) = -24x^2y^2 \)
3. \( 3(-2xy)(-2)^2 = 3(-2xy)(4) = -24xy \)
4. \( (-2)^3 = -8 \)

Thus, the expanded expression is:

\[
-8x^3y^3 - 24x^2y^2 - 24xy - 8
\]

Would you like further details on the expansion process?

Here are five related questions for further exploration:
1. What is the binomial expansion formula for higher powers like \( (a + b)^4 \)?
2. How does the sign of the terms change when dealing with negative numbers in binomials?
3. Can we apply a similar method to expand \( (-2xy + 3)^3 \)?
4. How does binomial expansion relate to Pascal's triangle?
5. How can the expanded form be factored back into the original expression?

**Tip:** When expanding binomials with negative terms, carefully track signs to avoid errors in the middle terms.

Expand and simplify the binomial expression (-2xy - 2)^3 using the binomial theorem. This step-by-step guide covers the application of algebraic concepts and the binomial expansion formula, suitable for high school students (Grades 9-12).

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