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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]

Firstly, let's focus on the inner expression [tex](4x^4 - 3x^2)^2[/tex]. We will expand this using the formula for the square of a binomial, [tex](a - b)^2 = a^2 - 2ab + b^2[/tex]:

[tex](4x^4 - 3x^2)^2 = (4x^4)^2 - 2 \times 4x^4 \times 3x^2 + (3x^2)^2[/tex].
Calculate each term separately:
- [tex](4x^4)^2 = 16x^8[/tex]
- [tex]-2 \times 4x^4 \times 3x^2 = -24x^6[/tex]
- [tex](3x^2)^2 = 9x^4[/tex]
Substitute these back into the expanded form:
- [tex]16x^8 - 24x^6 + 9x^4[/tex]

Next, combine this result with the rest of the given expression [tex]9x^5 + 3x[/tex]. Multiply [tex]3x[/tex] by each term in [tex]16x^8 - 24x^6 + 9x^4[/tex]:

[tex]3x \times 16x^8 = 48x^9[/tex]
[tex]3x \times (-24x^6) = -72x^7[/tex]
[tex]3x \times 9x^4 = 27x^5[/tex]

Now, add these results with the [tex]9x^5[/tex] from the original expression:

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

Simplify by combining like terms:

[tex]9x^5 + 27x^5 = 36x^5[/tex]

Thus, the simplified expression is:

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

Comparing this with the given options, the correct choice is:

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