To find the missing term in the product of the polynomials [tex]\( (-9m^6 + 12m^5) \)[/tex] and [tex]\( (m^2 + 2m + 1) \)[/tex], we need to perform polynomial multiplication and identify the coefficient of [tex]\( m^6 \)[/tex] in the resultant expression. Let's break it down step-by-step.
Given:
[tex]\[ (-9m^6 + 12m^5)(m^2 + 2m + 1) = -9m^8 - 6m^7 + ? + 12m^5 \][/tex]
1. Multiply [tex]\( -9m^6 \)[/tex] by each term in [tex]\( (m^2 + 2m + 1) \)[/tex]:
- [tex]\( -9m^6 \cdot m^2 = -9m^8 \)[/tex]
- [tex]\( -9m^6 \cdot 2m = -18m^7 \)[/tex]
- [tex]\( -9m^6 \cdot 1 = -9m^6 \)[/tex]
2. Multiply [tex]\( 12m^5 \)[/tex] by each term in [tex]\( (m^2 + 2m + 1) \)[/tex]:
- [tex]\( 12m^5 \cdot m^2 = 12m^7 \)[/tex]
- [tex]\( 12m^5 \cdot 2m = 24m^6 \)[/tex]
- [tex]\( 12m^5 \cdot 1 = 12m^5 \)[/tex]
3. Combine like terms:
- The [tex]\( m^8 \)[/tex]-term: [tex]\( -9m^8 \)[/tex]
- The [tex]\( m^7 \)[/tex]-term: [tex]\( -18m^7 + 12m^7 = -6m^7 \)[/tex]
- The [tex]\( m^6 \)[/tex]-term: [tex]\( -9m^6 + 24m^6 = 15m^6 \)[/tex]
- The [tex]\( m^5 \)[/tex]-term: [tex]\( 12m^5 \)[/tex]
Therefore, the product of [tex]\(( -9m^6 + 12m^5) (m^2 + 2m + 1)\)[/tex] is:
[tex]\[ -9m^8 - 6m^7 + 15m^6 + 12m^5 \][/tex]
Thus, the missing term in the expression [tex]\( -9m^8 - 6m^7 + ? + 12m^5 \)[/tex] is:
[tex]\[ 15m^6 \][/tex]
So, the correct answer is:
[tex]\[ 15m^6 \][/tex]
