To find the product of the polynomials \((3a^2b^4)\) and \((-8ab^3)\), we will follow these steps:
1. Multiply the coefficients:
[tex]\[
3 \cdot (-8) = -24
\][/tex]
2. Multiply the powers of \(a\):
- The term \(3a^2b^4\) has \(a^2\).
- The term \(-8ab^3\) has \(a\).
- When we multiply these, we add the exponents of \(a\):
[tex]\[
a^2 \cdot a^1 = a^{2+1} = a^3
\][/tex]
3. Multiply the powers of \(b\):
- The term \(3a^2b^4\) has \(b^4\).
- The term \(-8ab^3\) has \(b^3\).
- When we multiply these, we add the exponents of \(b\):
[tex]\[
b^4 \cdot b^3 = b^{4+3} = b^7
\][/tex]
Combining everything, the product of the given polynomials is:
[tex]\[
-24a^3b^7
\][/tex]
Thus, the correct answer is:
[tex]\[
-24a^3b^7
\][/tex]
