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Let's go through the questions step by step:

### Problem 1
We are given the following functions:
- \( g(x) = -2x^2 + 7x - 6 \)
- \( f(x) = 7x^3 - 3x^2 + 4x - 6 \)

**Part 1: Find \( (f + g)(x) \)**

To find \( (f + g)(x) \), we add the two functions:
\[
(f + g)(x) = f(x) + g(x) = (7x^3 - 3x^2 + 4x - 6) + (-2x^2 + 7x - 6)
\]
Combine like terms:
\[
(f + g)(x) = 7x^3 - 3x^2 - 2x^2 + 4x + 7x - 6 - 6
\]
\[
(f + g)(x) = 7x^3 - 5x^2 + 11x - 12
\]

**Part 2: Find \( (f - g)(1) \)**

Now we compute \( (f - g)(x) \), which is the difference of the two functions, evaluated at \( x = 1 \):
\[
(f - g)(x) = f(x) - g(x) = (7x^3 - 3x^2 + 4x - 6) - (-2x^2 + 7x - 6)
\]
Simplify:
\[
(f - g)(x) = 7x^3 - 3x^2 + 4x - 6 + 2x^2 - 7x + 6
\]
\[
(f - g)(x) = 7x^3 - x^2 - 3x
\]
Now evaluate this at \( x = 1 \):
\[
(f - g)(1) = 7(1)^3 - (1)^2 - 3(1) = 7 - 1 - 3 = 3
\]

### Problem 2
We are given:
- \( f(x) = x^3 - 216 \)
- \( g(x) = x - 6 \)

**Part 1: Find \( (f \cdot g)(-2) \)**

To find \( (f \cdot g)(x) \), multiply the two functions:
\[
(f \cdot g)(x) = f(x) \cdot g(x) = (x^3 - 216)(x - 6)
\]
First, expand this expression:
\[
(f \cdot g)(x) = x^3(x - 6) - 216(x - 6)
\]
\[
(f \cdot g)(x) = x^4 - 6x^3 - 216x + 1296
\]
Now evaluate at \( x = -2 \):
\[
(f \cdot g)(-2) = (-2)^4 - 6(-2)^3 - 216(-2) + 1296
\]
\[
(f \cdot g)(-2) = 16 + 6(8) + 432 + 1296 = 16 + 48 + 432 + 1296 = 1792
\]

**Part 2: Find \( \left( \frac{f}{g} \right)(x) \)**

To find \( \left( \frac{f}{g} \right)(x) \), divide the two functions:
\[
\left( \frac{f}{g} \right)(x) = \frac{x^3 - 216}{x - 6}
\]
We can simplify this by factoring the numerator using the difference of cubes:
\[
x^3 - 216 = (x - 6)(x^2 + 6x + 36)
\]
So,
\[
\left( \frac{f}{g} \right)(x) = \frac{(x - 6)(x^2 + 6x + 36)}{x - 6}
\]
For \( x \neq 6 \), the \( (x - 6) \) terms cancel, and we get:
\[
\left( \frac{f}{g} \right)(x) = x^2 + 6x + 36
\]

---

Would you like any further clarifications or details on these steps? Here are some related questions:

1. How would you calculate \( (f + g)(2) \) using the functions from Problem 1?
2. What happens if \( x = 6 \) in \( \left( \frac{f}{g} \right)(x) \)?
3. How do you factor the difference of cubes in general?
4. Can you derive the general formula for the product of two functions?
5. How would the approach change if \( f(x) \) and \( g(x) \) were exponential functions?

**Tip:** When simplifying rational expressions, always check if terms can cancel after factoring!

Given two functions, find operations such as addition, subtraction, multiplication, and division of the functions at specific points.

Learn how to perform operations on polynomial functions, including addition, subtraction, multiplication, and division. This problem involves applying algebraic techniques such as factorization and evaluating at specific points. Suitable for Grades 10-12 students.

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