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In the image provided, the task involves finding the quotient of two functions \( f(x) \) and \( g(x) \) and then simplifying the result. The functions given are:

\[
f(x) = 4x + 20
\]
\[
g(x) = x^2 - 3x - 40
\]

### Part (a) - Find \( \frac{f(x)}{g(x)} \) and simplify the result.

We begin by forming the quotient:

\[
\frac{f(x)}{g(x)} = \frac{4x + 20}{x^2 - 3x - 40}
\]

Now, let's simplify this expression:

1. **Factor the numerator** \( 4x + 20 \):
   \[
   4x + 20 = 4(x + 5)
   \]

2. **Factor the denominator** \( x^2 - 3x - 40 \):
   To factor this, find two numbers that multiply to \( -40 \) and add to \( -3 \). These numbers are \( -8 \) and \( 5 \):
   \[
   x^2 - 3x - 40 = (x - 8)(x + 5)
   \]

Thus, the quotient becomes:

\[
\frac{4(x + 5)}{(x - 8)(x + 5)}
\]

3. **Cancel out common factors** \( (x + 5) \) in the numerator and the denominator:
   \[
   \frac{4(x + 5)}{(x - 8)(x + 5)} = \frac{4}{x - 8}, \quad \text{for} \, x \neq -5
   \]

The simplified result is:
\[
\frac{f(x)}{g(x)} = \frac{4}{x - 8}, \quad \text{for} \, x \neq -5
\]

### Part (b) - State the domain of \( \frac{f(x)}{g(x)} \)

The domain of the quotient function is determined by the restrictions that make the denominator zero. From the expression \( g(x) = (x - 8)(x + 5) \), the denominator equals zero at \( x = 8 \) and \( x = -5 \).

Therefore, the domain excludes these two values. In set-builder notation, the domain is:

\[
\{ x \in \mathbb{R} \mid x \neq -5, \, x \neq 8 \}
\]

Would you like further details on any part of this solution?

### 5 Related Questions:
1. How do you find the domain of a rational function?
2. What are the steps to factor quadratic expressions?
3. How do you simplify complex rational expressions?
4. What happens if there are no common factors to cancel in a rational function?
5. How do you verify the cancellation of factors in a rational function?

**Tip:** Always check for values that make the denominator zero before simplifying rational expressions, as these values are excluded from the domain.

Consider the functions f(x) = 4x + 20 and g(x) = x^2 - 3x - 40. Find the quotient f(x)/g(x), simplify the result, and state the domain of f(x)/g(x).

Learn how to simplify the quotient of the functions f(x) = 4x + 20 and g(x) = x^2 - 3x - 40. This problem covers key algebraic concepts, including factoring and simplifying rational expressions. Suitable for students in Grades 9-11.

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