Question

Construct a table to find the missing term in the product (−9m^6+12m^5)(m^2+2m+1)=−9m^8−6m^7+?+12m^5.

Answer 1

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
1. Multiply \(-9m^6\) with each term in \((m^2 + 2m + 1)\):
- \(-9m^6 \cdot m^2 = -9m^8\)
- \(-9m^6 \cdot 2m = -18m^7\)
- \(-9m^6 \cdot 1 = -9m^6\)

2. 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:
\[
\boxed{15m^6}
\]