To find the missing term in the product \((-9m^6 + 12m^5)(m^2 + 2m + 1)\), we can first expand the product using the distributive property (FOIL method).
Step 1: Expand using distribution
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Multiply \(-9m^6\) with 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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Multiply \(12m^5\) with each term in \((m^2 + 2m + 1)\):
- \(12m^5 \cdot m^2 = 12m^7\)
- \(12m^5 \cdot 2m = 24m^6\)
- \(12m^5 \cdot 1 = 12m^5\)
Step 2: Combine all terms
Now, combine all the results from the multiplications: \[ -9m^8 + (-18m^7 + 12m^7) + (-9m^6 + 24m^6) + 12m^5 \]
Step 3: Simplify the expression
Combine the like terms:
- The \(m^8\) term: \(-9m^8\)
- The \(m^7\) terms: \(-18m^7 + 12m^7 = -6m^7\)
- The \(m^6\) terms: \(-9m^6 + 24m^6 = 15m^6\)
- The \(m^5\) term: \(12m^5\)
Putting it all together, we have: \[ -9m^8 - 6m^7 + 15m^6 + 12m^5 \]
Result Formulation
Now, according to the original expression provided in the question: \[ (-9m^6 + 12m^5)(m^2 + 2m + 1) = -9m^8 - 6m^7 + ? + 12m^5 \] We identified the terms:
- The term for \(m^6\) is \(15m^6\).
Conclusion
Thus, the missing term in the product is: \[ \boxed{15m^6} \]