Let's expand the given polynomial expression step-by-step:
We have:
[tex]\[
(2x^5 + 3y^4)(-4x^2 + 9y^4)
\][/tex]
Using the distributive property (also known as the FOIL method for binomials), we must multiply each term in the first polynomial by each term in the second polynomial.
1. Multiply [tex]\(2x^5\)[/tex] by [tex]\(-4x^2\)[/tex]:
[tex]\[
2x^5 \cdot (-4x^2) = -8x^{5+2} = -8x^7
\][/tex]
2. Multiply [tex]\(2x^5\)[/tex] by [tex]\(9y^4\)[/tex]:
[tex]\[
2x^5 \cdot 9y^4 = 18x^5y^4
\][/tex]
3. Multiply [tex]\(3y^4\)[/tex] by [tex]\(-4x^2\)[/tex]:
[tex]\[
3y^4 \cdot (-4x^2) = -12x^2y^4
\][/tex]
4. Multiply [tex]\(3y^4\)[/tex] by [tex]\(9y^4\)[/tex]:
[tex]\[
3y^4 \cdot 9y^4 = 27y^{4+4} = 27y^8
\][/tex]
Now, let's sum up all these terms:
[tex]\[
-8x^7 + 18x^5y^4 - 12x^2y^4 + 27y^8
\][/tex]
Thus, the correct expression is:
[tex]\[
-8x^7 + 18x^5y^4 - 12x^2y^4 + 27y^8
\][/tex]
Comparing this with the provided options, the correct answer is:
C. [tex]\(-8x^7 + 18x^5y^4 - 12x^2y^4 + 27y^8\)[/tex]
