To properly solve the problem \( 6x^2y^4(5x^2 - 3x^2y^2 + 4y^2) \), we will distribute \( 6x^2y^4 \) to each term inside the parentheses. Let's go through this step-by-step.
1. Distribute \( 6x^2y^4 \) to \( 5x^2 \):
[tex]\[
6x^2y^4 \cdot 5x^2 = 6 \cdot 5 \cdot x^2 \cdot x^2 \cdot y^4 = 30x^4y^4
\][/tex]
2. Distribute \( 6x^2y^4 \) to \(-3x^2y^2\):
[tex]\[
6x^2y^4 \cdot (-3x^2y^2) = 6 \cdot (-3) \cdot x^2 \cdot x^2 \cdot y^4 \cdot y^2 = -18x^4y^6
\][/tex]
3. Distribute \( 6x^2y^4 \) to \( 4y^2 \):
[tex]\[
6x^2y^4 \cdot 4y^2 = 6 \cdot 4 \cdot x^2 \cdot y^4 \cdot y^2 = 24x^2y^6
\][/tex]
Now, combine the results:
[tex]\[
30x^4y^4 - 18x^4y^6 + 24x^2y^6
\][/tex]
Therefore, the multiplied expression simplifies to:
[tex]\[
30x^4y^4 - 18x^4y^6 + 24x^2y^6
\][/tex]
Among the given options, the correct answer is:
[tex]\[
\boxed{30x^4y^4 - 18x^4y^6 + 24x^2y^6}
\][/tex]
