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------------------------------------------------ Which expression is equivalent to [tex]$9x^5 + 3x(4x^4 - 3x^2)^2$[/tex]?

A. [tex]$48x^9 - 24x^6 + 9x^5 + 9x^4$[/tex]

B. [tex][tex]$48x^9 + 9x^5 - 9x^4$[/tex][/tex]

C. [tex]$48x^9 + 36x^5$[/tex]

D. [tex]$48x^9 - 72x^7 + 36x^5$[/tex]

To simplify the given expression [tex]$ 9x^5 + 3x(4x^4 - 3x^2)^2 $[/tex] and determine which of the provided options it is equivalent to, we can break down the problem into smaller parts:

1. Simplify the Inner Expression:
The expression inside the parentheses is [tex]$ 4x^4 - 3x^2 $[/tex].

2. Square the Inner Expression:
We want to find [tex]$(4x^4 - 3x^2)^2$[/tex].
[tex]\[
(4x^4 - 3x^2)^2 = (4x^4 - 3x^2)(4x^4 - 3x^2)
\][/tex]
Use the distributive property (FOIL method):
[tex]\[
(4x^4)(4x^4) + (4x^4)(-3x^2) + (-3x^2)(4x^4) + (-3x^2)(-3x^2)
\][/tex]
Simplify each term:
- [tex]$ (4x^4)(4x^4) = 16x^8 $[/tex]
- [tex]$ (4x^4)(-3x^2) = -12x^6 $[/tex]
- [tex]$ (-3x^2)(4x^4) = -12x^6 $[/tex]
- [tex]$ (-3x^2)(-3x^2) = 9x^4 $[/tex]

Combine like terms:
[tex]\[
16x^8 - 24x^6 + 9x^4
\][/tex]

3. Multiply by [tex]$ 3x $[/tex]:
Next, multiply the squared expression [tex]$ 16x^8 - 24x^6 + 9x^4 $[/tex] by [tex]$ 3x $[/tex]:
[tex]\[
3x(16x^8 - 24x^6 + 9x^4)
\][/tex]
Distribute [tex]$ 3x $[/tex]:
- [tex]$ 3x \cdot 16x^8 = 48x^9 $[/tex]
- [tex]$ 3x \cdot (-24x^6) = -72x^7 $[/tex]
- [tex]$ 3x \cdot 9x^4 = 27x^5 $[/tex]

This gives us:
[tex]\[
48x^9 - 72x^7 + 27x^5
\][/tex]

4. Add [tex]$ 9x^5 $[/tex] to This Result:
Add [tex]$ 9x^5 $[/tex] from the original expression:
[tex]\[
48x^9 - 72x^7 + 27x^5 + 9x^5 = 48x^9 - 72x^7 + 36x^5
\][/tex]

The expanded expression is [tex]$ 48x^9 - 72x^7 + 36x^5 $[/tex].

Therefore, the expression is equivalent to the option:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]