To find the product of the expressions \((2q^9 + 3q^7)(-6q^2 + 9)\), we can use the distributive property (also known as the FOIL method for binomials).
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Distribute \(2q^9\):
- \(2q^9 \cdot (-6q^2) = -12q^{11}\)
- \(2q^9 \cdot 9 = 18q^9\)
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Distribute \(3q^7\):
- \(3q^7 \cdot (-6q^2) = -18q^{9}\)
- \(3q^7 \cdot 9 = 27q^7\)
Now we combine all the terms: \[ -12q^{11} + 18q^9 - 18q^9 + 27q^7 \]
The \(18q^9\) and \(-18q^9\) cancel each other out: \[ -12q^{11} + 27q^{7} \]
Thus, the final product is: \[ -12q^{11} + 27q^{7} \]
So the correct response is: \(-12q^{11} + 27q^{7}\)