To find the product of the expressions \( (3q^9 + 3q^7) \) and \( (-6q^{-2} + 9) \), we can use the distributive property (also known as the FOIL method for binomials).
Let's multiply each term in the first expression by each term in the second expression:
- \( 3q^9 \cdot (-6q^{-2}) = -18q^{9 - 2} = -18q^7 \)
- \( 3q^9 \cdot 9 = 27q^9 \)
- \( 3q^7 \cdot (-6q^{-2}) = -18q^{7 - 2} = -18q^5 \)
- \( 3q^7 \cdot 9 = 27q^7 \)
Now, we combine all of these results together:
\[ -18q^7 + 27q^9 - 18q^5 + 27q^7 \]
Next, we'll combine the like terms:
- Combine \( -18q^7 \) and \( 27q^7 \): \[ -18q^7 + 27q^7 = 9q^7 \]
Putting it all together, we get:
\[ 27q^9 + 9q^7 - 18q^5 \]
Thus, the final product of \( (3q^9 + 3q^7)(-6q^{-2} + 9) \) is:
\[ 27q^9 + 9q^7 - 18q^5 \]
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