Question

Construct a table to find the missing term in the product (-9m^6 + 12m^5) (m² + 2m + 1) = -9m^8 - 6m^7 + ? + 12m^5.

33m^6

3m^6

15m^6

-9m^6

Answer 1

To find the missing term in the expression \((-9m^6 + 12m^5)(m^2 + 2m + 1) = -9m^8 - 6m^7 + ? + 12m^5\), we will start by expanding the left-hand side.

We expand \((-9m^6 + 12m^5)(m^2 + 2m + 1)\) step-by-step:

1. **Multiply \(-9m^6\) by each term in \((m^2 + 2m + 1)\)**:

- \(-9m^6 \cdot m^2 = -9m^8\)
- \(-9m^6 \cdot 2m = -18m^7\)
- \(-9m^6 \cdot 1 = -9m^6\)

This gives us:
\[
-9m^8 - 18m^7 - 9m^6
\]

2. **Multiply \(12m^5\) by each term in \((m^2 + 2m + 1)\)**:

- \(12m^5 \cdot m^2 = 12m^7\)
- \(12m^5 \cdot 2m = 24m^6\)
- \(12m^5 \cdot 1 = 12m^5\)

This gives us:
\[
12m^7 + 24m^6 + 12m^5
\]

3. **Combine all the terms from both expansions**:
\[
(-9m^8 - 18m^7 - 9m^6) + (12m^7 + 24m^6 + 12m^5)
\]

Now, we can combine like terms:
- Combine \(m^8\) terms: \(-9m^8\)
- Combine \(m^7\) terms: \(-18m^7 + 12m^7 = -6m^7\)
- Combine \(m^6\) terms: \(-9m^6 + 24m^6 = 15m^6\)
- The \(m^5\) term remains as \(12m^5\)

So, the final polynomial after expansion is:
\[
-9m^8 - 6m^7 + 15m^6 + 12m^5
\]

4. **Identify the missing term**:
From the expression \( -9m^8 - 6m^7 + ? + 12m^5\), we can see that the missing term is \(15m^6\).

Therefore, the answer to the missing term is:
\[
\boxed{15m^6}
\]