To determine which expression is equivalent to [tex]\(\left(x^2 - 3x \right) \left(4x^2 + 2x - 9\right)\)[/tex], we need to simplify each of the given options and check which one matches the expanded form.
First, let's expand and simplify [tex]\(\left(x^2 - 3x \right) \left(4x^2 + 2x - 9\right)\)[/tex]:
### Step 1: Expand [tex]\(\left(x^2 - 3x \right) \left(4x^2 + 2x - 9\right)\)[/tex]
[tex]\[
(x^2)(4x^2) + (x^2)(2x) + (x^2)(-9) - (3x)(4x^2) - (3x)(2x) - (3x)(-9)
\][/tex]
[tex]\[
= 4x^4 + 2x^3 - 9x^2 - 12x^3 - 6x^2 + 27x
\][/tex]
### Step 2: Combine like terms
[tex]\[
= 4x^4 + (2x^3 - 12x^3) + (-9x^2 - 6x^2) + 27x
\][/tex]
[tex]\[
= 4x^4 - 10x^3 - 15x^2 + 27x
\][/tex]
Now that we have the expanded form [tex]\(4x^4 - 10x^3 - 15x^2 + 27x\)[/tex], let's analyze each of the options to see which one is equivalent to this polynomial.
### Step 3: Check each option
#### Option A: [tex]\(x^2\left(4 x^2+2 x\right)-3 x(2 x-9)\)[/tex]
[tex]\[
x^2(4x^2 + 2x) - 3x(2x - 9)
\][/tex]
[tex]\[
= 4x^4 + 2x^3 - 6x^2 + 27x
\][/tex]
This does not match [tex]\(4x^4 - 10x^3 - 15x^2 + 27x\)[/tex].
#### Option B: [tex]\(x^2 \left(4 x^2 + 2 x - 9\right) + 3 x \left(4 x^2 + 2 x - 9\right)\)[/tex]
[tex]\[
x^2(4x^2 + 2x - 9) + 3x(4x^2 + 2x - 9)
\][/tex]
[tex]\[
= 4x^4 + 2x^3 - 9x^2 + 12x^3 + 6x^2 - 27x
\][/tex]
[tex]\[
= 4x^4 + 14x^3 - 3x^2 - 27x
\][/tex]
This does not match [tex]\(4x^4 - 10x^3 - 15x^2 + 27x\)[/tex].
#### Option C: [tex]\(x^2\left(4 x^2+2 x-9\right)-3x\left(4x^2+2x-9\right)\)[/tex]
[tex]\[
x^2(4x^2 + 2x - 9) - 3x(4x^2 + 2x - 9)
\][/tex]
[tex]\[
= 4x^4 + 2x^3 - 9x^2 - 12x^3 - 6x^2 + 27x
\][/tex]
[tex]\[
= 4x^4 - 10x^3 - 15x^2 + 27x
\][/tex]
This matches the expanded form exactly.
#### Option D: [tex]\(x^2\left(4x^2+2x-9\right)-3x\)[/tex]
[tex]\[
x^2(4x^2 + 2x - 9) - 3x
\][/tex]
[tex]\[
= 4x^4 + 2x^3 - 9x^2 - 3x
\][/tex]
This does not match [tex]\(4x^4 - 10x^3 - 15x^2 + 27x\)[/tex].
### Conclusion
The correct answer is [tex]\(\boxed{C}\)[/tex], as option C is the expression equivalent to [tex]\(\left(x^2 - 3x\right) \left(4x^2 + 2x - 9\right)\)[/tex].
