To find the equivalent expression for the polynomial [tex]\(\left(2 x^5 + 3 y^4\right)\left(-4 x^2 + 9 y^4\right)\)[/tex], we need to multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
Let's start by distributing each term:
[tex]\[
(2x^5)(-4x^2) + (2x^5)(9y^4) + (3y^4)(-4x^2) + (3y^4)(9y^4)
\][/tex]
Now, we'll perform each multiplication individually:
1. [tex]\((2x^5)(-4x^2) = -8x^{5+2} = -8x^7\)[/tex]
2. [tex]\((2x^5)(9y^4) = 18x^5y^4\)[/tex]
3. [tex]\((3y^4)(-4x^2) = -12x^2y^4\)[/tex]
4. [tex]\((3y^4)(9y^4) = 27y^{4+4} = 27y^8\)[/tex]
Now, combining these results:
[tex]\[
-8x^7 + 18x^5y^4 - 12x^2y^4 + 27y^8
\][/tex]
This expression matches one of the options provided. Therefore, the correct answer is:
[tex]\[
\boxed{-8 x^7+18 x^5 y^4-12 x^2 y^4+27 y^8}
\][/tex]
Hence, the correct answer is:
B. [tex]\(-8 x^7+18 x^5 y^4-12 x^2 y^4+27 y^8\)[/tex]
