To calculate the area bounded by the curves $f(x) = x^2 + 8x + 16$, $g(x) = 8x + 80$, and the x-axis over the interval $[-10, -4]$, we'll proceed as follows:

Step 1: Set up the area calculation

We need to determine the region where $g(x)$ is above $f(x)$ within the interval $[-10, -4]$.

1. Given functions:

   • $f(x) = x^2 + 8x + 16$
   • $g(x) = 8x + 80$

2. Interval:

   • The interval is $[-10, -4]$.

Step 2: Find the points of intersection of $f(x)$ and $g(x)$ in the interval

Set $f(x) = g(x)$ to find the points of intersection:

$x^2 + 8x + 16 = 8x + 80$

Simplify by subtracting $8x + 16$ from both sides:

$x^2 = 64$

Taking the square root of both sides:

$x = \pm 8$

Within the interval $[-10, -4]$, only $x = -8$ is relevant. So, we have intersections at $x = -10$ and $x = -8$ for this interval.

Step 3: Set up the integrals

To find the area, split it into two regions from $[-10, -8]$ and $[-8, -4]$:

1. For $[-10, -8]$, $g(x)$ is above $f(x)$.

   The area for this section is: $A_1 = \int_{-10}^{-8} \left( g(x) - f(x) \right) \, dx$

2. For $[-8, -4]$, $g(x)$ is still above $f(x)$.

   The area for this section is: $A_2 = \int_{-8}^{-4} \left( g(x) - f(x) \right) \, dx$

Step 4: Evaluate the integrals

Calculate $g(x) - f(x)$:

$g(x) - f(x) = (8x + 80) - (x^2 + 8x + 16)$ $= -x^2 + 64$

Now we integrate $-x^2 + 64$ over the two intervals:

1. For $A_1$: $A_1 = \int_{-10}^{-8} (-x^2 + 64) \, dx$

2. For $A_2$: $A_2 = \int_{-8}^{-4} (-x^2 + 64) \, dx$

Step 5: Compute the integrals

Let's evaluate these integrals:

1. Integral for $A_1$: $A_1 = \int_{-10}^{-8} (-x^2 + 64) \, dx = \left[ -\frac{x^3}{3} + 64x \right]_{-10}^{-8}$

2. Integral for $A_2$: $A_2 = \int_{-8}^{-4} (-x^2 + 64) \, dx = \left[ -\frac{x^3}{3} + 64x \right]_{-8}^{-4}$

After evaluating these, add $A_1$ and $A_2$ to get the total area.

Let me calculate this explicitly.The total area bounded by the curves $f(x) = x^2 + 8x + 16$, $g(x) = 8x + 80$, and the x-axis over the interval $[-10, -4]$ is $72$ square units.

Would you like further details on the calculations, or do you have any questions?

Here are a few related questions to explore further:

1. How would the area change if the interval was modified?
2. What if $f(x)$ and $g(x)$ had different intersections in a broader interval?
3. How does the shape of the parabolic function $f(x)$ affect the bounded area?
4. What are the implications if the x-axis was not considered as a boundary?
5. How can we visualize these functions and the area on a graph?

Tip: When dealing with areas bounded by curves, always check where one function crosses over another to determine the correct boundaries and intervals.