To find the missing term in the product \((-9m^6 + 12m^5)(m^2 + 2m + 1) = -9m^8 - 6m^7 + ? + 12m^5\), we can start by expanding the left-hand side of the equation.
Let's break it down:
-
First Term: \(-9m^6 \cdot m^2\) \[ -9m^6 \cdot m^2 = -9m^8 \]
-
Second Term: \(-9m^6 \cdot 2m\) \[ -9m^6 \cdot 2m = -18m^7 \]
-
Third Term: \(-9m^6 \cdot 1\) \[ -9m^6 \cdot 1 = -9m^6 \]
-
Fourth Term: \(12m^5 \cdot m^2\) \[ 12m^5 \cdot m^2 = 12m^7 \]
-
Fifth Term: \(12m^5 \cdot 2m\) \[ 12m^5 \cdot 2m = 24m^6 \]
-
Sixth Term: \(12m^5 \cdot 1\) \[ 12m^5 \cdot 1 = 12m^5 \]
Now, we can combine all these terms:
From the calculations:
- The \(m^8\) coefficient: \( -9m^8 \)
- The \(m^7\) coefficient: \( -18m^7 + 12m^7 = -6m^7 \)
- The \(m^6\) coefficient: \( -9m^6 + 24m^6 = 15m^6 \)
- The \(m^5\) coefficient: \( +12m^5 \)
Putting it all together:
\[ (-9m^8 - 6m^7 + 15m^6 + 12m^5) \]
Therefore, the missing term represented by \(?\) is \(15m^6\).
Thus, the correct answer is:
15m^6