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Let's break the question into parts and solve each of them systematically.

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### (a) **Quotient and remainder of the polynomial division:**

You are asked to divide \( x^3 - 2x^2 - 11x + c \) by \( x^2 + 3x + 4 \).

Let's perform polynomial long division.

Dividing the leading term \( x^3 \) by \( x^2 \), we get \( x \).

Now multiply \( x \) by \( x^2 + 3x + 4 \), which gives \( x^3 + 3x^2 + 4x \).

Now subtract \( (x^3 + 3x^2 + 4x) \) from \( (x^3 - 2x^2 - 11x + c) \):

\[
(x^3 - 2x^2 - 11x + c) - (x^3 + 3x^2 + 4x) = -5x^2 - 15x + c
\]

Now divide \( -5x^2 \) by \( x^2 \), which gives \( -5 \).

Multiply \( -5 \) by \( x^2 + 3x + 4 \), which gives \( -5x^2 - 15x - 20 \).

Now subtract \( (-5x^2 - 15x - 20) \) from \( (-5x^2 - 15x + c) \):

\[
(-5x^2 - 15x + c) - (-5x^2 - 15x - 20) = c + 20
\]

Thus, the quotient is \( x - 5 \) and the remainder is \( c + 20 \).

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### (b) **Determining if functions are even, odd, or neither:**

#### (i) \( f(x) = \frac{x^4 + 5}{1 - x} \)

- To check if it's even or odd, we compute \( f(-x) \):

\[
f(-x) = \frac{(-x)^4 + 5}{1 - (-x)} = \frac{x^4 + 5}{1 + x}
\]

Since \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \), the function is **neither even nor odd**.

#### (ii) \( g(x) = \sin(77x) \)

- Since \( \sin(-x) = -\sin(x) \), we have:

\[
g(-x) = \sin(77(-x)) = -\sin(77x) = -g(x)
\]

Thus, \( g(x) \) is an **odd function**.

#### (iii) \( h(x) = \cos^m(x) \), where \( m \) is a positive integer

- Since \( \cos(-x) = \cos(x) \), we have:

\[
h(-x) = \cos^m(-x) = \cos^m(x) = h(x)
\]

Thus, \( h(x) \) is an **even function**.

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### (c) **Sketching the graph of the rational function \( f(x) \):**

You are provided with the following information about \( f(x) \):

- **Domain**: \( (-\infty, -2) \cup (-2, \infty) \)
- **X-intercept**: \( x = -1 \)
- **Y-intercept**: \( y = -\frac{1}{2} \)
- **Increasing on**: \( (-\infty, -6) \), **Decreasing on**: \( (-6, -2) \) and \( (-2, \infty) \)
- **Local maximum** at \( (-6, -\frac{1}{4}) \)
- **Asymptotes**: Vertical asymptote at \( x = -2 \), Horizontal asymptote at \( y = -1 \)
- **Neither even nor odd**.

From this information, we can summarize the sketching procedure:

- The function has a vertical asymptote at \( x = -2 \), meaning the graph will approach this line but never cross it.
- The horizontal asymptote is at \( y = -1 \), so as \( x \to \infty \) or \( x \to -\infty \), the function will approach \( y = -1 \).
- The function is increasing on \( (-\infty, -6) \) and has a local maximum at \( (-6, -\frac{1}{4}) \). After that, it decreases on \( (-6, -2) \) and \( (-2, \infty) \).
- The x-intercept at \( x = -1 \) means the graph will cross the x-axis at this point.
- The y-intercept at \( y = -\frac{1}{2} \) means the graph will cross the y-axis at this point.

This gives us enough information to sketch the graph.

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Would you like further details or an explicit graph for part (c)? Let me know if you want me to delve deeper.

### 5 Related Questions:
1. How do you find the quotient and remainder in polynomial division?
2. What are the steps to check if a function is even or odd?
3. Can all trigonometric functions be categorized as even or odd?
4. How do vertical and horizontal asymptotes affect the shape of a rational function's graph?
5. What techniques can you use to determine increasing and decreasing intervals of a function?

### Tip:
When checking whether a function is even or odd, always remember to substitute \( -x \) in place of \( x \) and simplify!

Find the quotient and remainder on dividing x^3 - 2x^2 - 11x + c by x^2 + 3x + 4, where c is a real number. Determine whether each of the following functions are even, odd, or neither: (i) f(x) = (x^4 + 5)/(1 - x), (ii) g(x) = sin(77x), (iii) h(x) = cos^m(x), where m is a positive integer. Sketch the graph of a rational function f based on the provided characteristics.

Learn how to solve polynomial division, check if a function is even or odd, and sketch the graph of a rational function based on its key features. This detailed problem covers advanced algebra and graphing concepts, including asymptotes, x and y intercepts, and local maxima.

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