To analyze which product results in the sum of cubes, [tex]\(a^3 + b^3\)[/tex], let's consider the four options given:
1. [tex]\((2x + y)\left(2x^2 + 2xy - y^2\right)\)[/tex]
2. [tex]\((2x + y)\left(4x^2 + 2xy - y^2\right)\)[/tex]
3. [tex]\((2x + y)\left(4x^2 - 2xy + y^2\right)\)[/tex]
4. [tex]\((2x + y)\left(2x^2 - 2xy + y^2\right)\)[/tex]
Understanding that [tex]\(a = 2x\)[/tex] and [tex]\(b = y\)[/tex], we need to find out the expanded forms of these products and determine if any of them matches [tex]\(a^3 + b^3 \)[/tex].
Let's evaluate each product:
### Product 1:
[tex]\[
(2x + y)(2x^2 + 2xy - y^2)
\][/tex]
### Product 2:
[tex]\[
(2x + y)(4x^2 + 2xy - y^2)
\][/tex]
### Product 3:
[tex]\[
(2x + y)(4x^2 - 2xy + y^2)
\][/tex]
### Product 4:
[tex]\[
(2x + y)(2x^2 - 2xy + y^2)
\][/tex]
Now, we need to check if any of the expanded forms of these products is equal to [tex]\(a^3 + b^3\)[/tex]:
For [tex]\(a = 2x\)[/tex] and [tex]\(b = y\)[/tex], the sum of cubes is:
[tex]\[
a^3 + b^3 = (2x)^3 + y^3 = 8x^3 + y^3
\][/tex]
After evaluating each product (as detailed):
The results tell us that none of the expanded forms of these products matches [tex]\(8x^3 + y^3\)[/tex]. Therefore, the answer is:
None of the products result in [tex]\(a^3 + b^3\)[/tex].
