Sure, let's work through the process of using the Distributive Property to multiply the given polynomials step-by-step.
We start with the expression:
[tex]\[ 3x^2(2x^4 - 15x) \][/tex]
The Distributive Property states that [tex]\(a(b + c) = ab + ac\)[/tex]. In this case, we will distribute [tex]\(3x^2\)[/tex] to each term inside the parentheses.
1. Distribute [tex]\(3x^2\)[/tex] to [tex]\(2x^4\)[/tex]:
[tex]\[
3x^2 \cdot 2x^4 = 3 \cdot 2 \cdot x^2 \cdot x^4 = 6x^{2+4} = 6x^6
\][/tex]
2. Distribute [tex]\(3x^2\)[/tex] to [tex]\(-15x\)[/tex]:
[tex]\[
3x^2 \cdot (-15x) = 3 \cdot (-15) \cdot x^2 \cdot x = -45x^{2+1} = -45x^3
\][/tex]
Now, combine the results from each distribution step:
[tex]\[
6x^6 - 45x^3
\][/tex]
So, the final simplified expression after applying the Distributive Property is:
[tex]\[ 3x^2(2x^4 - 15x) = 6x^6 - 45x^3 \][/tex]
