To find the missing term in the product \((-9m^6 + 12m^5)(m^2 + 2m + 1)\), we can distribute each term from the first polynomial to each term in the second polynomial and collect like terms.
Step-by-step Distribution
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Distribute \(-9m^6\) to each term in \((m^2 + 2m + 1)\):
- \(-9m^6 \cdot m^2 = -9m^8\)
- \(-9m^6 \cdot 2m = -18m^7\)
- \(-9m^6 \cdot 1 = -9m^6\)
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Distribute \(12m^5\) to each term in \((m^2 + 2m + 1)\):
- \(12m^5 \cdot m^2 = 12m^7\)
- \(12m^5 \cdot 2m = 24m^6\)
- \(12m^5 \cdot 1 = 12m^5\)
Combine like terms
Now, we can combine the results from both distributions:
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The \(m^8\) terms: \[ -9m^8 \]
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The \(m^7\) terms: \[ -18m^7 + 12m^7 = -6m^7 \]
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The \(m^6\) terms: \[ -9m^6 + 24m^6 = 15m^6 \]
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The remaining \(m^5\) term: \[ 12m^5 \]
Resulting Product
Putting it all together, we have: \[ (-9m^6 + 12m^5)(m^2 + 2m + 1) = -9m^8 - 6m^7 + 15m^6 + 12m^5 \]
Thus, the missing term in the product is 15m^6.
Final Answer
15m^6
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