To multiply the polynomials \((6s^4 + 15s^3)(6s^3 - 15s^4)\), we can use a table to organize the multiplication of each term from the first polynomial with each term from the second polynomial.
Here’s how you can set up a multiplication table:
| | \(6s^3\) | \(-15s^4\) | |----------|----------|------------| | \(6s^4\) | \(36s^7\) | \(-90s^8\) | | \(15s^3\)| \(90s^6\) | \(-225s^7\)|
Now, let's fill in the table by multiplying the terms:
- \(6s^4 \cdot 6s^3 = 36s^7\)
- \(6s^4 \cdot (-15s^4) = -90s^8\)
- \(15s^3 \cdot 6s^3 = 90s^6\)
- \(15s^3 \cdot (-15s^4) = -225s^7\)
Next, let's combine the results from the table:
- From \(6s^4 \cdot (-15s^4)\): \(-90s^8\)
- From \(6s^4 \cdot 6s^3\): \(36s^7\)
- From \(15s^3 \cdot 6s^3\): \(90s^6\)
- From \(15s^3 \cdot (-15s^4)\): \(-225s^7\)
Now, we combine like terms:
- The \(s^8\) term: \(-90s^8\)
- The \(s^7\) terms: \(36s^7 - 225s^7 = -189s^7\)
- The \(s^6\) term: \(90s^6\)
Now, adding these up gives us:
\[ -90s^8 - 189s^7 + 90s^6 \]
Hence, the resulting polynomial is:
\[ -90s^8 - 189s^7 + 90s^6 \]
Among the options provided, the correct response is:
\(-90s^8 - 189s^7 + 90s^6\)