What is the product of (x^4)(3x^3 - 2)(4x^2 + 5x)?

1. Distribute the first two terms: Multiply $x^4$ by $(3x^3 - 2)$. This is done by multiplying $x^4$ by each term inside the parentheses: $x^4 \times 3x^3 = 3x^{4+3} = 3x^7$ and $x^4 \times -2 = -2x^4$. So, the expression becomes $(3x^7 - 2x^4)$.

2. Multiply the resulting binomials: Now, multiply the result from Step 1, $(3x^7 - 2x^4)$, by the third term, $(4x^2 + 5x)$. We will use the distributive property (or FOIL method) for binomials. Multiply each term in the first binomial by each term in the second binomial:
- $(3x^7) \times (4x^2) = 12x^{7+2} = 12x^9$
- $(3x^7) \times (5x) = 15x^{7+1} = 15x^8$
- $(-2x^4) \times (4x^2) = -8x^{4+2} = -8x^6$
- $(-2x^4) \times (5x) = -10x^{4+1} = -10x^5$

1. Combine the terms: Add all the products from Step 2: $12x^9 + 15x^8 - 8x^6 - 10x^5$. However, looking at the options, it seems there might be a typo in the problem or the options provided. Let's re-examine the multiplication in step 2, specifically the term $(-2x^4) \times (4x^2)$. The exponent should be $4+2=6$, resulting in $-8x^6$. If we assume the problem meant to have terms that would lead to one of the options, let's re-evaluate. Let's assume the problem is exactly as written and check the options again. It seems there might be a mistake in my calculation or the provided options. Let's redo step 2 carefully.

2. Multiply the resulting binomials (re-evaluation): Multiply $(3x^7 - 2x^4)$ by $(4x^2 + 5x)$.
- $(3x^7) \times (4x^2) = 12x^{7+2} = 12x^9$
- $(3x^7) \times (5x) = 15x^{7+1} = 15x^8$
- $(-2x^4) \times (4x^2) = -8x^{4+2} = -8x^6$
- $(-2x^4) \times (5x) = -10x^{4+1} = -10x^5$

Summing these gives: $12x^9 + 15x^8 - 8x^6 - 10x^5$. This result is not among the options. Let's check the options again. The options are:
- $12x^{24} + 15x^{12} - 8x^8 - 10x^4$
- $12x^{24} - 10x^4$
- $12x^9 - 10x^5$
- $12x^9 + 15x^8 - 8x^6 - 10x^5$

It appears there was a typo in my transcription of the options. The last option provided in the image is $12x^9 + 15x^8 - 8x^6 - 10x^5$. Let me re-examine the original problem and the options provided in the image to ensure I am not making a mistake.

Looking at the image again, the options are:
1. $12x^{24} + 15x^{12} - 8x^8 - 10x^4$
2. $12x^{24} - 10x^4$
3. $12x^9 - 10x^5$
4. $12x^9 + 15x^8 - 8x^6 - 10x^5$

My calculation yielded $12x^9 + 15x^8 - 8x^6 - 10x^5$. This matches the fourth option. Let me re-verify the calculation of the term $-8x^6$. It comes from $(-2x^4) \times (4x^2) = -8x^{4+2} = -8x^6$. This is correct.

Let me check if I made any mistake in the first step. $x^4 (3x^3 - 2) = 3x^7 - 2x^4$. This is correct.

Let me check the multiplication of $(3x^7 - 2x^4)(4x^2 + 5x)$ again.
$(3x^7)(4x^2) = 12x^9$
$(3x^7)(5x) = 15x^8$
$(-2x^4)(4x^2) = -8x^6$
$(-2x^4)(5x) = -10x^5$

Summing these: $12x^9 + 15x^8 - 8x^6 - 10x^5$. This is indeed the correct result.

It seems there might be a typo in the provided options in the image, as the term $-8x^6$ is present in my calculation but the option shows $-8x^8$. Let me assume there is a typo in the problem and it should have been $x^6$ instead of $x^8$ in the option, or a typo in the original expression. However, I must work with the given problem and options.

Let me re-read the problem and options very carefully from the image.
Problem: $(x^4)(3x^3 - 2)(4x^2 + 5x)$?
Options:
1. $12x^{24} + 15x^{12} - 8x^8 - 10x^4$
2. $12x^{24} - 10x^4$
3. $12x^9 - 10x^5$
4. $12x^9 + 15x^8 - 8x^6 - 10x^5$

My calculation is $12x^9 + 15x^8 - 8x^6 - 10x^5$. This matches the fourth option exactly. I misread the fourth option in my previous check.

1. Final Answer: The product of the given expression is $12x^9 + 15x^8 - 8x^6 - 10x^5$.