To determine which expression is equivalent to the given polynomial expression [tex]\((2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4)\)[/tex], we can expand the product step by step.
Consider the terms in the first polynomial: [tex]\(2x^2\)[/tex] and [tex]\(-3y^2\)[/tex].
We need to multiply each term in the first polynomial by each term in the second polynomial [tex]\((4x^4 + 6x^2y^2 + 9y^4)\)[/tex].
1. Multiply [tex]\(2x^2\)[/tex] by each term:
[tex]\[
2x^2 \cdot 4x^4 = 8x^6
\][/tex]
[tex]\[
2x^2 \cdot 6x^2y^2 = 12x^4y^2
\][/tex]
[tex]\[
2x^2 \cdot 9y^4 = 18x^2y^4
\][/tex]
2. Multiply [tex]\(-3y^2\)[/tex] by each term:
[tex]\[
-3y^2 \cdot 4x^4 = -12x^4y^2
\][/tex]
[tex]\[
-3y^2 \cdot 6x^2y^2 = -18x^2y^4
\][/tex]
[tex]\[
-3y^2 \cdot 9y^4 = -27y^6
\][/tex]
Now, add all the terms together:
[tex]\[
8x^6 + 12x^4y^2 + 18x^2y^4 - 12x^4y^2 - 18x^2y^4 - 27y^6
\][/tex]
Combine like terms:
[tex]\[
8x^6 + (12x^4y^2 - 12x^4y^2) + (18x^2y^4 - 18x^2y^4) - 27y^6
\][/tex]
Simplify:
[tex]\[
8x^6 - 27y^6
\][/tex]
Thus, the expression equivalent to [tex]\((2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4)\)[/tex] is:
[tex]\[
8x^6 - 27y^6
\][/tex]
Therefore, the correct answer is:
A. [tex]\(8x^6 - 27y^6\)[/tex]
