To determine the product of the given polynomials [tex]\((x^4)(3x^3 - 2)(4x^2 + 5x)\)[/tex], let's go through the solution step-by-step.
1. Expand [tex]\((3x^3 - 2)(4x^2 + 5x)\)[/tex]:
[tex]\[
(3x^3 - 2)(4x^2 + 5x) = (3x^3)(4x^2) + (3x^3)(5x) + (-2)(4x^2) + (-2)(5x)
\][/tex]
Simplify each term:
[tex]\[
= 12x^5 + 15x^4 - 8x^2 - 10x
\][/tex]
2. Now multiply the result by [tex]\(x^4\)[/tex]:
[tex]\[
x^4(12x^5 + 15x^4 - 8x^2 - 10x) = x^4 \cdot 12x^5 + x^4 \cdot 15x^4 + x^4 \cdot (-8x^2) + x^4 \cdot (-10x)
\][/tex]
Simplify each term:
[tex]\[
= 12x^9 + 15x^8 - 8x^6 - 10x^5
\][/tex]
Therefore, the product of the given polynomials [tex]\((x^4)(3x^3 - 2)(4x^2 + 5x)\)[/tex] is:
[tex]\[
12x^9 + 15x^8 - 8x^6 - 10x^5
\][/tex]
So, the correct answer is:
[tex]\[
12x^9 + 15x^8 - 8x^6 - 10x^5
\][/tex]
