To solve the problem of finding [tex]\((f \cdot g)(m)\)[/tex], we need to multiply the functions [tex]\(f(m)\)[/tex] and [tex]\(g(m)\)[/tex] together and evaluate the product.
Given the functions:
[tex]\[
f(m) = 5m^2 - m + 2
\][/tex]
[tex]\[
g(m) = -17m^4 - 15m^2
\][/tex]
We will determine the product [tex]\( (f \cdot g)(m) \)[/tex].
### Step-by-Step Solution:
1. Evaluate [tex]\(f(m)\)[/tex] at [tex]\(m = 1\)[/tex]:
[tex]\[
f(1) = 5(1)^2 - 1 + 2 = 5 - 1 + 2 = 6
\][/tex]
2. Evaluate [tex]\(g(m)\)[/tex] at [tex]\(m = 1\)[/tex]:
[tex]\[
g(1) = -17(1)^4 - 15(1)^2 = -17 - 15 = -32
\][/tex]
3. Compute the product [tex]\((f \cdot g)(1)\)[/tex]:
[tex]\[
(f \cdot g)(1) = f(1) \cdot g(1) = 6 \cdot (-32) = -192
\][/tex]
So, the value of [tex]\((f \cdot g)(m)\)[/tex] when [tex]\(m = 1\)[/tex] is [tex]\(-192\)[/tex].
### Hand-generated Expression:
Given the hand-generated expression:
[tex]\[
-85m^6 + 17m^5 - \frac{109}{m^4} + 15m^3 - 3
\][/tex]
Evaluate this expression at [tex]\(m = 1\)[/tex]:
1. Substitute [tex]\(m = 1\)[/tex] into the expression:
[tex]\[
-85(1)^6 + 17(1)^5 - \frac{109}{(1)^4} + 15(1)^3 - 3
\][/tex]
2. Simplify the terms:
[tex]\[
-85(1) + 17(1) - 109 + 15(1) - 3 = -85 + 17 - 109 + 15 - 3
\][/tex]
3. Simplify further by combining like terms:
[tex]\[
-85 + 17 - 109 + 15 - 3 = -165
\][/tex]
So, the value of the hand-generated expression when [tex]\(m = 1\)[/tex] is [tex]\(-165\)[/tex].
### Conclusion:
1. The product [tex]\((f \cdot g)(m)\)[/tex] when [tex]\(m = 1\)[/tex] is [tex]\(-192\)[/tex].
2. The hand-generated expression evaluates to [tex]\(-165\)[/tex] when [tex]\(m = 1\)[/tex].
Hence, the final answers are:
- [tex]\((f \cdot g)(1) = -192\)[/tex]
- Hand-generated result evaluated at [tex]\(m = 1\)[/tex] is [tex]\(-165\)[/tex]
