To solve the inequality \(2x + 1 \leq x^3 \leq 2x + 4\), we'll break it into two separate inequalities and solve each:

1. \(2x + 1 \leq x^3\)
2. \(x^3 \leq 2x + 4\)

### Step 1: Solve \(2x + 1 \leq x^3\)

This can be rewritten as:
\[
x^3 - 2x - 1 \geq 0
\]

To find the roots, we can try factoring or use numerical methods if factoring is complex. 

The function \(f(x) = x^3 - 2x - 1\) can be analyzed for roots.

### Step 2: Solve \(x^3 \leq 2x + 4\)

This can be rewritten as:
\[
x^3 - 2x - 4 \leq 0
\]

Again, we need to analyze the roots of the function \(g(x) = x^3 - 2x - 4\).

### Step 3: Combine the Results

Once we have the roots of these inequalities, we will determine the intervals where the inequalities hold true. These intervals will be the solution to the original compound inequality.

Would you like to see the full detailed solution including the exact roots or numerical approximations, and the final intervals where the compound inequality is satisfied?

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**Relative Questions:**

1. What is the importance of finding the roots when solving inequalities?
2. How can the derivative of a function help in solving inequalities like this one?
3. What is the significance of testing intervals when solving compound inequalities?
4. How would the solution change if the inequality were strict (e.g., \(2x + 1 < x^3 < 2x + 4\))?
5. Can inequalities like these have solutions involving imaginary numbers?

**Tip:** When solving inequalities involving polynomials, it's often helpful to first find the roots of the corresponding equation. This allows you to test intervals to see where the inequality holds.

Learn how to solve the compound inequality 2x + 1 <= x^3 <= 2x + 4 with a detailed step-by-step solution. This problem covers key mathematical concepts like inequalities, polynomial functions, and interval analysis. Suitable for advanced high school students (Grades 10-12).

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