Sure, let's break down the expression [tex]\((2x + 5y)^2 + (2x - 5y)^2\)[/tex] step by step to simplify it.
1. Expand each squared term:
[tex]\[
(2x + 5y)^2 = (2x + 5y)(2x + 5y)
\][/tex]
Using the distributive property (FOIL method for binomials):
[tex]\[
= 2x(2x) + 2x(5y) + 5y(2x) + 5y(5y)
= 4x^2 + 10xy + 10xy + 25y^2
= 4x^2 + 20xy + 25y^2
\][/tex]
2. Expand the second squared term:
[tex]\[
(2x - 5y)^2 = (2x - 5y)(2x - 5y)
\][/tex]
Using the distributive property (FOIL method for binomials):
[tex]\[
= 2x(2x) + 2x(-5y) + (-5y)(2x) + (-5y)(-5y)
= 4x^2 - 10xy - 10xy + 25y^2
= 4x^2 - 20xy + 25y^2
\][/tex]
3. Add the two expanded expressions together:
[tex]\[
(2x + 5y)^2 + (2x - 5y)^2
\][/tex]
Combine the results from the above expansions:
[tex]\[
= (4x^2 + 20xy + 25y^2) + (4x^2 - 20xy + 25y^2)
\][/tex]
4. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
4x^2 + 4x^2 = 8x^2
\][/tex]
- Combine the [tex]\(xy\)[/tex] terms:
[tex]\[
20xy - 20xy = 0
\][/tex]
- Combine the [tex]\(y^2\)[/tex] terms:
[tex]\[
25y^2 + 25y^2 = 50y^2
\][/tex]
5. Write the simplified expression:
[tex]\[
8x^2 + 50y^2
\][/tex]
Therefore, the simplified form of [tex]\((2x + 5y)^2 + (2x - 5y)^2\)[/tex] is:
[tex]\[
8x^2 + 50y^2
\][/tex]
