To multiply the polynomials \((6s^4 + 15s^3)(6s^3 - 15s^4)\) using a table, we can organize it as follows:
- List the terms of the first polynomial across the top.
- List the terms of the second polynomial down the side.
Here's how you can set it up:
\[ \begin{array}{c|c|c} & 6s^3 & -15s^4 \ \hline 6s^4 & 36s^7 & -90s^8 \ \hline 15s^3 & 90s^6 & -225s^7 \ \end{array} \]
Now we will compute each entry:
- \(6s^4 \cdot 6s^3 = 36s^7\)
- \(6s^4 \cdot (-15s^4) = -90s^8\)
- \(15s^3 \cdot 6s^3 = 90s^6\)
- \(15s^3 \cdot (-15s^4) = -225s^7\)
Now let's combine like terms from the results:
- For \(s^8\): \(-90s^8\)
- For \(s^7\): \(36s^7 - 225s^7 = -189s^7\)
- For \(s^6\): \(90s^6\)
Combining these, the final result is:
\[ -90s^8 - 189s^7 + 90s^6 \]
Thus, the answer is:
\(-90s^8 - 189s^7 + 90s^6\)