To find the product of [tex]\((4x^3 + 2x^2)(6x - 9)\)[/tex], we will use the distributive property to expand and simplify the expression step-by-step.
1. Distribute each term in the first polynomial [tex]\((4x^3 + 2x^2)\)[/tex] with each term in the second polynomial [tex]\((6x - 9)\)[/tex].
Let's start with [tex]\(4x^3\)[/tex]:
[tex]\[
4x^3 \cdot 6x = 24x^{3+1} = 24x^4
\][/tex]
[tex]\[
4x^3 \cdot (-9) = -36x^3
\][/tex]
Next, distribute [tex]\(2x^2\)[/tex]:
[tex]\[
2x^2 \cdot 6x = 12x^{2+1} = 12x^3
\][/tex]
[tex]\[
2x^2 \cdot (-9) = -18x^2
\][/tex]
2. Combine all the distributed terms together:
[tex]\[
24x^4 - 36x^3 + 12x^3 - 18x^2
\][/tex]
3. Combine like terms:
[tex]\[
24x^4 + (-36x^3 + 12x^3) - 18x^2
\][/tex]
[tex]\[
24x^4 - 24x^3 - 18x^2
\][/tex]
The final expanded and simplified expression is:
[tex]\[
(4x^3 + 2x^2)(6x - 9) = 24x^4 - 24x^3 - 18x^2
\][/tex]
So, [tex]\(\left(4 x^3+2 x^2\right)(6 x-9)= 24x^4 - 24x^3 - 18x^2\)[/tex].
