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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]$48x^9 + 9x^5 - 9x^4$[/tex]
C. [tex]$48x^9 + 36x^5$[/tex]
D. [tex]$48x^9 - 72x^7 + 36x^5$[/tex]

To solve the problem of finding an expression equivalent to [tex]$9x^5 + 3x(4x^4 - 3x^2)^2$[/tex], we need to work through the expression step by step to simplify it. Here's how it can be approached:

1. Start with the expression given:
[tex]\[
9x^5 + 3x(4x^4 - 3x^2)^2
\][/tex]

2. Expand the squared term [tex]$(4x^4 - 3x^2)^2$[/tex]:

To expand this, we can apply the formula [tex]$(a - b)^2 = a^2 - 2ab + b^2$[/tex].
- [tex]$a = 4x^4$[/tex] and [tex]$b = 3x^2$[/tex].
- [tex]$a^2 = (4x^4)^2 = 16x^8$[/tex].
- [tex]$b^2 = (3x^2)^2 = 9x^4$[/tex].
- [tex]$2ab = 2 \times (4x^4) \times (3x^2) = 24x^6$[/tex].

So, [tex]$(4x^4 - 3x^2)^2 = 16x^8 - 24x^6 + 9x^4$[/tex].

3. Multiply the expanded squared term by [tex]$3x$[/tex]:

[tex]\[
3x \times (16x^8 - 24x^6 + 9x^4)
\][/tex]

Distribute [tex]$3x$[/tex] across each term in the expanded squared term:
- [tex]$3x \times 16x^8 = 48x^9$[/tex]
- [tex]$3x \times (-24x^6) = -72x^7$[/tex]
- [tex]$3x \times 9x^4 = 27x^5$[/tex]

4. Combine all terms:

Combine this result with the original [tex]$9x^5$[/tex]:

[tex]\[
9x^5 + 48x^9 - 72x^7 + 27x^5
\][/tex]

Combine like terms ([tex]$9x^5 + 27x^5$[/tex]):

[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]

5. Identify the equivalent expression:

The simplified expression is:
[tex]\[
48x^9 - 72x^7 + 36x^5
\][/tex]

Thus, the expression equivalent to [tex]$9x^5 + 3x(4x^4 - 3x^2)^2$[/tex] is:

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

This matches the option:
- [tex]$48 x^9 - 72 x^7 + 36 x^5$[/tex]