Sure, let's solve each part of the problem step by step based on the provided functions:
First, let's define our functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f(x) = 5x^2 + 3x - 4 \][/tex]
[tex]\[ g(x) = -2x + 8 \][/tex]
### Part 1: [tex]\( \frac{F(x)}{g(x)} \)[/tex]
To find [tex]\( \frac{F(x)}{g(x)} \)[/tex], we note that [tex]\( F(x) \)[/tex] is the same as [tex]\( f(x) \)[/tex].
[tex]\[ \frac{F(x)}{g(x)} = \frac{f(x)}{g(x)} \][/tex]
### Part 2: [tex]\( \frac{f(x)}{g(x)} \)[/tex]
Again, this is simply:
[tex]\[ \frac{f(x)}{g(x)} = \frac{f(x)}{g(x)} \][/tex]
### Part 3: [tex]\( f(2) \times (g(3))^2 \)[/tex]
We need to evaluate the functions at specific points:
1. [tex]\( f(2) \)[/tex]:
[tex]\[ f(2) = 5(2)^2 + 3(2) - 4 = 5(4) + 6 - 4 = 20 + 6 - 4 = 22 \][/tex]
2. [tex]\( g(3) \)[/tex]:
[tex]\[ g(3) = -2(3) + 8 = -6 + 8 = 2 \][/tex]
Now, we need to calculate [tex]\( f(2) \times (g(3))^2 \)[/tex]:
[tex]\[ f(2) \times (g(3))^2 = 22 \times 2^2 = 22 \times 4 = 88 \][/tex]
### Part 4: [tex]\( \frac{f(-3)}{g(2)} \)[/tex]
Evaluate the functions at these points:
1. [tex]\( f(-3) \)[/tex]:
[tex]\[ f(-3) = 5(-3)^2 + 3(-3) - 4 = 5(9) - 9 - 4 = 45 - 9 - 4 = 32 \][/tex]
2. [tex]\( g(2) \)[/tex]:
[tex]\[ g(2) = -2(2) + 8 = -4 + 8 = 4 \][/tex]
Now, we need to calculate [tex]\( \frac{f(-3)}{g(2)} \)[/tex]:
[tex]\[ \frac{f(-3)}{g(2)} = \frac{32}{4} = 8 \][/tex]
### Summary of Results:
1. [tex]\( \frac{F(x)}{g(x)} = \frac{f(x)}{g(x)} \)[/tex]
2. [tex]\( \frac{f(x)}{g(x)} = \frac{f(x)}{g(x)} \)[/tex]
3. [tex]\( f(2) \times (g(3))^2 = 88 \)[/tex]
4. [tex]\( \frac{f(-3)}{g(2)} = 8 \)[/tex]
