Find the product of [tex]\left(4 x^3 + 2 x^2\right)(6 x - 9)[/tex]. Provide your answer in descending order of exponents.

[tex]
\left(4 x^3 + 2 x^2\right)(6 x - 9) =
[/tex]

[tex]\boxed{}[/tex]



Answer :

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].