To find the product of the two polynomials [tex]\((-2m^3 + 3m^2 - m)(4m^2 + m - 5)\)[/tex], we need to multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
Let's denote the polynomials as:
[tex]\[ P(m) = -2m^3 + 3m^2 - m \][/tex]
[tex]\[ Q(m) = 4m^2 + m - 5 \][/tex]
We will multiply each term in [tex]\( P(m) \)[/tex] by each term in [tex]\( Q(m) \)[/tex]:
1. Multiply [tex]\(-2m^3\)[/tex] by each term in [tex]\( Q(m) \)[/tex]:
[tex]\[
-2m^3 \cdot 4m^2 = -8m^5
\][/tex]
[tex]\[
-2m^3 \cdot m = -2m^4
\][/tex]
[tex]\[
-2m^3 \cdot (-5) = 10m^3
\][/tex]
2. Multiply [tex]\( 3m^2 \)[/tex] by each term in [tex]\( Q(m) \)[/tex]:
[tex]\[
3m^2 \cdot 4m^2 = 12m^4
\][/tex]
[tex]\[
3m^2 \cdot m = 3m^3
\][/tex]
[tex]\[
3m^2 \cdot (-5) = -15m^2
\][/tex]
3. Multiply [tex]\( -m \)[/tex] by each term in [tex]\( Q(m) \)[/tex]:
[tex]\[
-m \cdot 4m^2 = -4m^3
\][/tex]
[tex]\[
-m \cdot m = -m^2
\][/tex]
[tex]\[
-m \cdot (-5) = 5m
\][/tex]
Now, we combine all these terms:
[tex]\[
-8m^5 - 2m^4 + 10m^3 + 12m^4 + 3m^3 - 15m^2 - 4m^3 - m^2 + 5m
\][/tex]
Next, combine like terms:
[tex]\[
\text{Combine } m^5 \text{ terms: } -8m^5
\][/tex]
[tex]\[
\text{Combine } m^4 \text{ terms: } (-2m^4 + 12m^4 = 10m^4)
\][/tex]
[tex]\[
\text{Combine } m^3 \text{ terms: } (10m^3 + 3m^3 - 4m^3 = 9m^3)
\][/tex]
[tex]\[
\text{Combine } m^2 \text{ terms: } (-15m^2 - m^2 = -16m^2)
\][/tex]
[tex]\[
\text{Combine } m \text{ terms: } 5m
\][/tex]
Therefore, the standard form polynomial that represents the product is:
[tex]\[
-8m^5 + 10m^4 + 9m^3 - 16m^2 + 5m
\][/tex]
