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------------------------------------------------ What is the product?

[tex](4x)(-3x^8)(-7x^3)[/tex]

A. [tex]-84x^{12}[/tex]
B. [tex]-84x^{24}[/tex]
C. [tex]84x^{12}[/tex]
D. [tex]84x^{24}[/tex]

To find the product of the expression [tex]$(4x)(-3x^8)(-7x^3)$[/tex], we need to follow these steps:

1. Multiply the coefficients: Look at the numbers (without the variables) first. We have:
- [tex]$4$[/tex]
- [tex]$-3$[/tex]
- [tex]$-7$[/tex]

Multiply these numbers together:
[tex]\[
4 \times (-3) \times (-7)
\][/tex]
Multiplying two negative numbers gives a positive product. Therefore:
[tex]\[
4 \times (-3) = -12 \quad \Rightarrow \quad (-12) \times (-7) = 84
\][/tex]

2. Add the exponents: Now, consider the exponents of [tex]$x$[/tex]. When you multiply powers with the same base, you add their exponents. So, we have:
- [tex]$x$[/tex] is the same as [tex]$x^1$[/tex]
- [tex]$x^8$[/tex]
- [tex]$x^3$[/tex]

Add these exponents together:
[tex]\[
1 + 8 + 3 = 12
\][/tex]

3. Combine the results: Plug in the calculated coefficient and the combined exponent into the expression to get the final product:
[tex]\[
84x^{12}
\][/tex]

Therefore, the product of [tex]$(4x)(-3x^8)(-7x^3)$[/tex] is [tex]$84x^{12}$[/tex]. The correct answer is [tex]$ 84x^{12} $[/tex].