We need to find the average rate of change of the function $f(x) = \frac{x-5}{x+3}$ from $x = -2$ to $x = 4$.

1. Problem Description

We need to find the average rate of change of the function $f(x) = \frac{x-5}{x+3}$ from $x = -2$ to $x = 4$.

2. Solution Steps

The average rate of change of a function $f(x)$ from $x = a$ to $x = b$ is given by the formula:

$\frac{f(b) - f(a)}{b - a}$

In this case, $f(x) = \frac{x-5}{x+3}$, $a = -2$, and $b = 4$.

First, we need to find $f(4)$ and $f(-2)$.

$f(4) = \frac{4-5}{4+3} = \frac{-1}{7}$

$f(-2) = \frac{-2-5}{-2+3} = \frac{-7}{1} = -7$

Now, we can find the average rate of change:

$\frac{f(4) - f(-2)}{4 - (-2)} = \frac{-\frac{1}{7} - (-7)}{4 + 2} = \frac{-\frac{1}{7} + 7}{6} = \frac{-\frac{1}{7} + \frac{49}{7}}{6} = \frac{\frac{48}{7}}{6} = \frac{48}{7} \cdot \frac{1}{6} = \frac{48}{42} = \frac{8}{7}$

3. Final Answer

The average rate of change is $\frac{8}{7}$.