To find the product of the polynomials:
[tex]\[
\left(6m^2 - 5m + 4\right) \left(-4m^2 - 3m - 7\right),
\][/tex]
we will use the distributive property, also known as the FOIL method (First, Outer, Inner, Last), to multiply each term in the first polynomial by each term in the second polynomial. Let us break this down step-by-step:
1. Multiply the first terms:
[tex]\[
6m^2 \cdot -4m^2 = -24m^4
\][/tex]
2. Multiply the outer terms:
[tex]\[
6m^2 \cdot -3m = -18m^3
\][/tex]
3. Multiply the next outer terms:
[tex]\[
6m^2 \cdot -7 = -42m^2
\][/tex]
4. Multiply the next outer terms:
[tex]\[
-5m \cdot -4m^2 = 20m^3
\][/tex]
5. Multiply the inner terms:
[tex]\[
-5m \cdot -3m = 15m^2
\][/tex]
6. Multiply the next inner terms:
[tex]\[
-5m \cdot -7 = 35m
\][/tex]
7. Multiply the last terms:
[tex]\[
4 \cdot -4m^2 = -16m^2
\][/tex]
8. Multiply the next last terms:
[tex]\[
4 \cdot -3m = -12m
\][/tex]
9. Multiply the last terms:
[tex]\[
4 \cdot -7 = -28
\][/tex]
We now sum all these results together:
[tex]\[
-24m^4 + (-18m^3) + (-42m^2) + 20m^3 + 15m^2 + 35m + (-16m^2) + (-12m) + (-28)
\][/tex]
Next, we combine like terms:
- The [tex]\(m^4\)[/tex] term: [tex]\(-24m^4\)[/tex]
- The [tex]\(m^3\)[/tex] terms: [tex]\(-18m^3 + 20m^3 = 2m^3\)[/tex]
- The [tex]\(m^2\)[/tex] terms: [tex]\(-42m^2 + 15m^2 - 16m^2 = -43m^2\)[/tex]
- The [tex]\(m\)[/tex] terms: [tex]\(35m - 12m = 23m\)[/tex]
- The constant term: [tex]\(-28\)[/tex]
Putting all these together, we get:
[tex]\[
-24m^4 + 2m^3 - 43m^2 + 23m - 28
\][/tex]
Thus, the product of the given polynomials is:
[tex]\[
-24m^4 + 2m^3 - 43m^2 + 23m - 28
\][/tex]
