To multiply the polynomials \((6s^4 + 15s^3)(6s^3 - 15s^4)\) using a table, we can construct a grid that represents the products of the terms in each polynomial.
Step 1: Set up the table
We will create a table with the terms from each polynomial. This means we will multiply each term from the first polynomial by each term from the second polynomial.
First polynomial: \(6s^4\) and \(15s^3\)
Second polynomial: \(6s^3\) and \(-15s^4\)
We can arrange them in a table:
\[ \begin{array}{c|c|c} & 6s^3 & -15s^4 \ \hline 6s^4 & 6s^4 \cdot 6s^3 & 6s^4 \cdot (-15s^4) \ \hline 15s^3 & 15s^3 \cdot 6s^3 & 15s^3 \cdot (-15s^4) \ \end{array} \]
Step 2: Fill in the table
Now we will compute each cell of the table:
- \(6s^4 \cdot 6s^3 = 36s^7\)
- \(6s^4 \cdot (-15s^4) = -90s^8\)
- \(15s^3 \cdot 6s^3 = 90s^6\)
- \(15s^3 \cdot (-15s^4) = -225s^7\)
So the completed table looks like this:
\[ \begin{array}{c|c|c} & 6s^3 & -15s^4 \ \hline 6s^4 & 36s^7 & -90s^8 \ \hline 15s^3 & 90s^6 & -225s^7 \ \end{array} \]
Step 3: Combine the results
Now we need to combine all the results from the table:
- Combine like terms:
- The \(s^8\) term: \(-90s^8\)
- The \(s^7\) terms: \(36s^7 - 225s^7 = -189s^7\)
- The \(s^6\) term: \(90s^6\)
Final Result
Putting it all together, we have:
\[ -90s^8 - 189s^7 + 90s^6 \]
So the final answer for the product of the polynomials \((6s^4 + 15s^3)(6s^3 - 15s^4)\) is:
\[ \boxed{-90s^8 - 189s^7 + 90s^6} \]