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The problem asks us to find the area of the region enclosed by the graphs of the functions:

\[
f(x) = x^2 + 2 \quad \text{and} \quad g(x) = 2x + 5
\]

### Steps to solve:
1. **Find the points of intersection** of the two curves by setting \( f(x) = g(x) \):
   \[
   x^2 + 2 = 2x + 5
   \]
   Simplifying this equation:
   \[
   x^2 - 2x - 3 = 0
   \]
   Factor the quadratic equation:
   \[
   (x - 3)(x + 1) = 0
   \]
   So, the points of intersection are \( x = 3 \) and \( x = -1 \).

2. **Set up the integral** to find the area between the curves. The area is given by the integral of the difference between the functions from \( x = -1 \) to \( x = 3 \):
   \[
   \text{Area} = \int_{-1}^{3} \left[ (2x + 5) - (x^2 + 2) \right] dx
   \]
   Simplify the expression inside the integral:
   \[
   \text{Area} = \int_{-1}^{3} \left( 2x + 5 - x^2 - 2 \right) dx = \int_{-1}^{3} \left( -x^2 + 2x + 3 \right) dx
   \]

3. **Compute the integral**:
   \[
   \int (-x^2 + 2x + 3) dx = -\frac{x^3}{3} + x^2 + 3x
   \]
   Evaluate this from \( x = -1 \) to \( x = 3 \):
   \[
   \left[ -\frac{(3)^3}{3} + (3)^2 + 3(3) \right] - \left[ -\frac{(-1)^3}{3} + (-1)^2 + 3(-1) \right]
   \]
   Simplifying both parts:
   \[
   \left[ -\frac{27}{3} + 9 + 9 \right] - \left[ \frac{1}{3} + 1 - 3 \right]
   \]
   \[
   \left[ -9 + 9 + 9 \right] - \left[ \frac{1}{3} - 2 \right]
   \]
   \[
   9 - \left( \frac{1}{3} - 2 \right) = 9 - \left( -\frac{5}{3} \right) = 9 + \frac{5}{3}
   \]
   Convert \( 9 \) to thirds:
   \[
   9 + \frac{5}{3} = \frac{27}{3} + \frac{5}{3} = \frac{32}{3}
   \]

Thus, the area of the region enclosed by the curves is \( \frac{32}{3} \).

Would you like a detailed explanation of any part of the solution, or have any further questions?

### 5 Related Questions:
1. How do you determine which function to subtract when finding the area between curves?
2. What would happen if the curves intersected at more points?
3. Can this method be applied if the curves enclose multiple regions?
4. How would you calculate the area if one of the functions was a trigonometric function?
5. What are the implications if the area is negative in such a calculation?

### Tip:
Always sketch the curves to visually confirm which function is on top to set up the correct integral bounds.

Find the area of the region enclosed by the graphs of f(x) = x^2 + 2 and g(x) = 2x + 5.

This problem involves finding the area enclosed by the curves f(x) = x^2 + 2 and g(x) = 2x + 5 using integration. It covers concepts like finding intersection points and applying the area formula for curves, and is suitable for high school calculus students.

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