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To find the volume of the solid, we need to first understand the problem. The solid has a base bounded by the functions \( p(x) \) and \( g(x) \), and slices perpendicular to the base are squares. The volume can be calculated by integrating the area of these squares along the x-axis.

Given:
- \( p(x) = 2x^2 + 11x + 26 \)
- \( g(x) = \frac{x^2}{2} + 26 \)

The side length of each square slice is the difference between \( p(x) \) and \( g(x) \), so the area of each square is:

\[
\text{Area} = \left( p(x) - g(x) \right)^2
\]

The volume \( V \) of the solid is then given by the integral of this area over the interval where the functions \( p(x) \) and \( g(x) \) intersect:

\[
V = \int_{a}^{b} \left( p(x) - g(x) \right)^2 dx
\]

### Step 1: Find the intersection points
To find the intersection points, set \( p(x) = g(x) \):

\[
2x^2 + 11x + 26 = \frac{x^2}{2} + 26
\]

Simplifying this equation:

\[
2x^2 - \frac{x^2}{2} + 11x = 0
\]

\[
\frac{4x^2 - x^2}{2} + 11x = 0
\]

\[
\frac{3x^2}{2} + 11x = 0
\]

Factor the equation:

\[
x \left( \frac{3x}{2} + 11 \right) = 0
\]

This gives \( x = 0 \) or \( x = -\frac{22}{3} \).

### Step 2: Set up the integral
The volume integral is:

\[
V = \int_{-\frac{22}{3}}^{0} \left( \left(2x^2 + 11x + 26 \right) - \left(\frac{x^2}{2} + 26 \right) \right)^2 dx
\]

Simplifying the expression inside the integral:

\[
V = \int_{-\frac{22}{3}}^{0} \left( \frac{3x^2}{2} + 11x \right)^2 dx
\]

### Step 3: Evaluate the integral
I'll now compute the integral to find the volume.

Let's calculate.The exact volume of the solid is \(\frac{644204}{405}\).

To express this as a decimal, we can calculate:

\[
V \approx 1590.1358
\]

So, the volume of the solid is approximately **1590.1358 cubic units**.

Would you like a detailed explanation or have any further questions?

Here are some related questions you might find interesting:

1. How would the volume change if the slices perpendicular to the base were equilateral triangles instead of squares?
2. How do you determine the intersection points of \( p(x) \) and \( g(x) \) for other functions?
3. What is the geometric significance of the integral of the area of squares?
4. How would the volume calculation change if the functions were different?
5. What if the base were bounded by three different functions instead of two?
6. How do you generalize the method for solids with other cross-sectional shapes?
7. What happens if the region between the curves is rotated about the x-axis instead?
8. How would the volume be affected if the functions were multiplied by a constant?

**Tip:** When calculating volumes of solids with known cross-sectional areas, always ensure that the limits of integration correctly reflect the boundaries of the region.

Learn how to calculate the volume of a solid bounded by two quadratic functions, using integration and the difference of squares. This advanced high school math problem covers integration techniques, quadratic equations, and geometric applications.

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