To solve the expression [tex]\((2x + \sqrt{2}y)^2\)[/tex], we need to expand it using algebraic rules. Here are the steps to expand the given expression:
1. Identify the Binomial Expansion Formula: The expression [tex]\((a + b)^2\)[/tex] expands to [tex]\(a^2 + 2ab + b^2\)[/tex].
2. Assign Values:
- Let [tex]\(a = 2x\)[/tex]
- Let [tex]\(b = \sqrt{2}y\)[/tex]
3. Apply the Binomial Expansion Formula:
- Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = (2x)^2 = 4x^2
\][/tex]
- Calculate [tex]\(2ab\)[/tex]:
[tex]\[
2ab = 2 \cdot (2x) \cdot (\sqrt{2}y) = 4x\sqrt{2}y
\][/tex]
- Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = (\sqrt{2}y)^2 = 2y^2
\][/tex]
4. Combine All Terms:
Putting it all together, we get:
[tex]\[
(2x + \sqrt{2}y)^2 = a^2 + 2ab + b^2 = 4x^2 + 4\sqrt{2}xy + 2y^2
\][/tex]
So, the expanded form of the given expression [tex]\( (2x + \sqrt{2}y)^2 \)[/tex] is:
[tex]\[ 4x^2 + 4\sqrt{2}xy + 2y^2 \][/tex]
