To determine the product of the expressions \((x^4)\), \((3x^3 - 2)\), and \((4 x^2 + 5 x)\), we will follow these steps:
1. Multiply the first two expressions:
Consider the first two expressions, \(x^4\) and \(3x^3 - 2\):
[tex]\[
x^4 \cdot (3x^3 - 2) = x^4 \cdot 3x^3 - x^4 \cdot 2 = 3x^7 - 2x^4
\][/tex]
2. Multiply the result with the third expression:
Now, multiply the expression obtained in step 1, which is \(3x^7 - 2x^4\), by \(4x^2 + 5x\):
Consider the product \( (3x^7 - 2x^4)(4x^2 + 5x) \). We will use the distributive property:
[tex]\[
\begin{align*}
(3x^7 - 2x^4)(4x^2 + 5x) &= 3x^7 \cdot 4x^2 + 3x^7 \cdot 5x - 2x^4 \cdot 4x^2 - 2x^4 \cdot 5x \\
&= 3x^7 \cdot 4x^2 + 3x^7 \cdot 5x - 2x^4 \cdot 4x^2 - 2x^4 \cdot 5x \\
&= 12x^9 + 15x^8 - 8x^6 - 10x^5
\end{align*}
\][/tex]
3. Compare the result with the given options:
The resulting polynomial from the multiplication is:
[tex]\[
12 x^9 + 15 x^8 - 8 x^6 - 10 x^5
\][/tex]
4. Identify the correct option:
- The first choice is \(12 x^9 + 15 x^8 - 8 x^6 - 10 x^5\), which exactly matches our computed result.
- The other choices are: \(12 x^{24} + 15 x^{12} - 8 x^8 - 10 x^4\), \(12 x^9 - 10 x^5\), and \(12 x^{24} - 10 x^4\). None of these match our result.
Therefore, the correct answer to the question is:
[tex]\[
12 x^9 + 15 x^8 - 8 x^6 - 10 x^5
\][/tex]
Hence, the correct option is the first one.
