To find the missing term in the expression \((−9m^6 + 12m^5)(m^2 + 2m + 1)\), we can distribute each term in the first polynomial across each term in the second polynomial.
Let's perform the multiplication step-by-step:
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Distribute \(-9m^6\) across the second polynomial \((m^2 + 2m + 1)\): \[ -9m^6 \cdot m^2 = -9m^8 \] \[ -9m^6 \cdot 2m = -18m^7 \] \[ -9m^6 \cdot 1 = -9m^6 \]
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Distribute \(12m^5\) across the second polynomial \((m^2 + 2m + 1)\): \[ 12m^5 \cdot m^2 = 12m^7 \] \[ 12m^5 \cdot 2m = 24m^6 \] \[ 12m^5 \cdot 1 = 12m^5 \]
Now we can combine all the terms together: \[ (-9m^8) + (-18m^7 + 12m^7) + (-9m^6 + 24m^6) + (12m^5) \]
Combine the like terms:
- For \(m^8\): \(-9m^8\)
- For \(m^7\): \(-18m^7 + 12m^7 = -6m^7\)
- For \(m^6\): \(-9m^6 + 24m^6 = 15m^6\)
- For \(m^5\): \(12m^5\)
Now we can put this all together: \[ -9m^8 - 6m^7 + 15m^6 + 12m^5 \]
Thus, based on the calculated product, we can see that the missing term in the expression \(−9m^8−6m^7+?+12m^5\) is:
\[ \boxed{15m^6} \]
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