To multiply the polynomials \( (6s^4 + 15s^3)(6s^3 - 15s^4) \), we can construct a table for the multiplication:
Step 1: Set up the table
We will multiply each term in the first polynomial by each term in the second polynomial.
- First Polynomial: \( 6s^4, 15s^3 \)
- Second Polynomial: \( 6s^3, -15s^4 \)
Step 2: Create the table
| | 6s³ | -15s⁴ | |----------|---------------|---------------| | 6s⁴ | \( 36s^7 \) | \(-90s^8\) | | 15s³ | \( 90s^6 \) | \(-225s^7\) |
Step 3: Collect the results
Now let's combine these results:
- From \( 6s^4 \times 6s^3 \): \( 36s^7 \)
- From \( 6s^4 \times -15s^4 \): \(-90s^8\)
- From \( 15s^3 \times 6s^3 \): \( 90s^6 \)
- From \( 15s^3 \times -15s^4 \): \(-225s^7\)
Step 4: Combine like terms
Now combine the 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 correct answer is:
\[ \boxed{-90s^8 - 189s^7 + 90s^6} \]