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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]

Answer :

Sure! Let's simplify the expression [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex] step by step.

### Step 1: Simplify the expression inside the parentheses

The expression inside the parentheses is [tex]\((4x^4 - 3x^2)^2\)[/tex]. We need to expand this using the formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].

[tex]\[ a = 4x^4 \quad \text{and} \quad b = 3x^2 \][/tex]

Applying the formula, we have:

[tex]\[
(4x^4 - 3x^2)^2 = (4x^4)^2 - 2(4x^4)(3x^2) + (3x^2)^2
\][/tex]

Calculating each term separately, we get:

- [tex]\((4x^4)^2 = 16x^8\)[/tex]

- [tex]\(2(4x^4)(3x^2) = 24x^6\)[/tex]

- [tex]\((3x^2)^2 = 9x^4\)[/tex]

Therefore, the expanded form of [tex]\((4x^4 - 3x^2)^2\)[/tex] is:

[tex]\[
16x^8 - 24x^6 + 9x^4
\][/tex]

### Step 2: Multiply by [tex]\(3x\)[/tex]

Now, multiply the expanded expression by [tex]\(3x\)[/tex]:

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

Calculate each term:

- [tex]\(3x \cdot 16x^8 = 48x^9\)[/tex]

- [tex]\(3x \cdot 24x^6 = 72x^7\)[/tex]

- [tex]\(3x \cdot 9x^4 = 27x^5\)[/tex]

So, multiplying gives:

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

### Step 3: Add the original term [tex]\(9x^5\)[/tex]

Add [tex]\(9x^5\)[/tex] to the expression:

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

Combine the like terms [tex]\(27x^5\)[/tex] and [tex]\(9x^5\)[/tex]:

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

Thus, the final expression is:

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

### Conclusion

The expression [tex]\(9x^5 + 3x(4x^4 - 3x^2)^2\)[/tex] simplifies to:

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

This matches option: [tex]\(48x^9 - 72x^7 + 36x^5\)[/tex].

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