To simplify the expression \((2x + 2y)^2\), we'll apply the binomial expansion formula:

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

In this case, \(a = 2x\) and \(b = 2y\).

Expanding the expression:

\[
(2x + 2y)^2 = (2x)^2 + 2 \cdot (2x) \cdot (2y) + (2y)^2
\]

Now, let's simplify each term:

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

Putting it all together:

\[
(2x + 2y)^2 = 4x^2 + 8xy + 4y^2
\]

So, the simplified expression is:

\[
4x^2 + 8xy + 4y^2
\]

Would you like further details or have any questions? Here are some related questions to explore:

1. How would this expansion change if the expression were \((2x - 2y)^2\)?
2. Can you factor the resulting expression \(4x^2 + 8xy + 4y^2\)?
3. What happens if \(x = y\)? How does the expression simplify?
4. How does the expression behave if \(x\) or \(y\) is set to zero?
5. How can you verify the expansion by substituting specific values for \(x\) and \(y\)?

**Tip:** When expanding binomials, it’s helpful to remember the basic binomial formula \((a + b)^2 = a^2 + 2ab + b^2\) to quickly expand and simplify.

Learn how to simplify the expression (2x + 2y)^2 using the binomial expansion formula. This problem involves expanding and simplifying binomials, specifically (2x + 2y)^2, and covers key algebraic concepts and formulas. Suitable for students in Grades 9-12.

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