The problem asks to determine the average rate of change of the function $f(x) = 4(x+2)^2 + 1$ on the interval $-1 < x < 3$.

1. Problem Description

The problem asks to determine the average rate of change of the function $f(x) = 4(x+2)^2 + 1$ on the interval $-1 < x < 3$.

2. Solution Steps

The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by the formula:

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

In this problem, we have $f(x) = 4(x+2)^2 + 1$, $a = -1$, and $b = 3$.

First, we evaluate $f(a) = f(-1)$:

$f(-1) = 4(-1+2)^2 + 1 = 4(1)^2 + 1 = 4(1) + 1 = 4 + 1 = 5$.

Next, we evaluate $f(b) = f(3)$:

$f(3) = 4(3+2)^2 + 1 = 4(5)^2 + 1 = 4(25) + 1 = 100 + 1 = 101$.

Now, we compute the average rate of change:

$\frac{f(3) - f(-1)}{3 - (-1)} = \frac{101 - 5}{3 - (-1)} = \frac{96}{4} = 24$.

3. Final Answer

The average rate of change of the function $f(x) = 4(x+2)^2 + 1$ on the interval $-1 < x < 3$ is 24.