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### (a) Deriving \( f'(x) \):
The function is given as:

\[
f(x) = \frac{x^2 + 4x - 12}{x + 6}
\]

This is a rational function, so to find \( f'(x) \), we will apply the **quotient rule**, which is:

\[
\left( \frac{u(x)}{v(x)} \right)' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]

Here, 
- \( u(x) = x^2 + 4x - 12 \)
- \( v(x) = x + 6 \)

Now, let's differentiate both:

1. \( u'(x) = 2x + 4 \)
2. \( v'(x) = 1 \)

Applying the quotient rule:

\[
f'(x) = \frac{(2x + 4)(x + 6) - (x^2 + 4x - 12)(1)}{(x + 6)^2}
\]

Simplifying the numerator:

\[
= (2x + 4)(x + 6) - (x^2 + 4x - 12)
\]
\[
= (2x^2 + 12x + 4x + 24) - (x^2 + 4x - 12)
\]
\[
= 2x^2 + 16x + 24 - x^2 - 4x + 12
\]
\[
= x^2 + 12x + 36
\]

So the derivative simplifies to:

\[
f'(x) = \frac{x^2 + 12x + 36}{(x + 6)^2}
\]

### (b) Finding the points where the tangent line is parallel to \( 100y - 100x = 11 \):

To find where the tangent line is parallel to the given line, we need to determine the slope of this line. The equation can be rewritten as:

\[
y = x + \frac{11}{100}
\]

Thus, the slope of this line is **1**. For the tangent to be parallel, we need \( f'(x) = 1 \).

We already know that:

\[
f'(x) = \frac{x^2 + 12x + 36}{(x + 6)^2}
\]

Set \( f'(x) = 1 \) and solve for \( x \):

\[
\frac{x^2 + 12x + 36}{(x + 6)^2} = 1
\]

Multiply both sides by \( (x + 6)^2 \):

\[
x^2 + 12x + 36 = (x + 6)^2
\]

Expanding the right-hand side:

\[
x^2 + 12x + 36 = x^2 + 12x + 36
\]

Since both sides are identical, it shows that the equation is true for all \( x \neq -6 \) (as \( x = -6 \) would make the denominator zero). So, every point on the curve has a slope of 1 except at \( x = -6 \).

#### Determining the points:
Now, we find the left-most and right-most points where the tangent line is parallel. The left-most and right-most points are likely to be at the limits of the domain of \( f(x) \), where \( x \neq -6 \). Evaluating the curve to find these values:

Left-most point: \( (x, y) = (-6, f(-6)) \)

Right-most point: \( (x, y) = \) based on behavior or further constraints.

Do you want more details on solving for these specific points?

### Expanding Concepts:
Here are five related questions to expand understanding:
1. How do you use the quotient rule to differentiate other rational functions?
2. What are the conditions for a function’s tangent line to be parallel to a given line?
3. How do you handle undefined points in rational functions, such as vertical asymptotes?
4. Can the slope of a tangent line be any constant, and how would the solution change?
5. How would you find horizontal or vertical asymptotes of this rational function?

### Tip:
Always simplify rational derivatives as much as possible before solving for specific conditions like slope or tangent lines.

Consider the function f(x) = (x^2 + 4x - 12) / (x + 6). Find f'(x) and all points P(x, y) on the curve where the tangent line is parallel to the line 100y - 100x = 11. Also, find the left-most and right-most points.

Learn how to apply the quotient rule to find the derivative of a rational function and solve for points on the curve where the tangent is parallel to a given line. This problem covers key calculus concepts, including differentiation, tangent slopes, and parallel line conditions.

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