Let's solve the problem step by step to find the correct answer.
We are given the polynomial expression:
[tex]\[
(2x^2 - 3y^2)(4x^4 + 6x^2y^2 + 9y^4)
\][/tex]
Our goal is to expand this expression and match it with one of the given options. To do this, let's distribute each term in the first polynomial by each term in the second polynomial.
1. Start with [tex]\(2x^2\)[/tex] and distribute:
[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. Now distribute [tex]\(-3y^2\)[/tex]:
[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]
3. Now, let's combine all the terms obtained:
[tex]\[
8x^6 + 12x^4y^2 + 18x^2y^4 - 12x^4y^2 - 18x^2y^4 - 27y^6
\][/tex]
4. Simplify the expression by combining like terms:
[tex]\[
8x^6 + (12x^4y^2 - 12x^4y^2) + (18x^2y^4 - 18x^2y^4) - 27y^6
\][/tex]
All intermediate terms cancel out:
[tex]\[
8x^6 - 27y^6
\][/tex]
Thus, the expanded expression is:
[tex]\[
8x^6 - 27y^6
\][/tex]
This matches option D:
[tex]\[
\boxed{8x^6 - 27y^6}
\][/tex]
Therefore, the correct answer is option D.
