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The problem in the image reads:

"The equation \( P(x) = x^4 + x^3 + 2x^2 + x + 2 \) has at least one root in the interval \((-1, 1)\). True or False?"

To determine whether this statement is true or false, we can analyze the function \( P(x) \) on the interval \((-1, 1)\).

### Steps:
1. **Evaluate \( P(x) \) at \( x = -1 \) and \( x = 1 \):**
   - \( P(-1) = (-1)^4 + (-1)^3 + 2(-1)^2 + (-1) + 2 \)
   - \( P(-1) = 1 - 1 + 2 - 1 + 2 = 3 \)
   - \( P(1) = 1^4 + 1^3 + 2(1)^2 + 1 + 2 \)
   - \( P(1) = 1 + 1 + 2 + 1 + 2 = 7 \)

   Both \( P(-1) \) and \( P(1) \) are positive.

2. **Check the behavior of \( P(x) \) on the interval \( (-1, 1) \):**
   - The function \( P(x) \) is a polynomial and therefore continuous.
   - Since \( P(-1) > 0 \) and \( P(1) > 0 \), if \( P(x) \) remains positive throughout the interval \( (-1, 1) \), it won't have any roots in that interval.

3. **Intermediate Value Theorem (IVT) Analysis:**
   - For the polynomial \( P(x) \) to have a root in the interval \( (-1, 1) \), the function must change sign somewhere in this interval (i.e., \( P(x) \) should be negative at some point if it is positive at the endpoints).
   - However, since we calculated \( P(x) \) at the endpoints and both are positive, we need to investigate further by checking whether \( P(x) \) can turn negative within the interval.

   - To do that, let's try evaluating \( P(x) \) at a few points within \( (-1, 1) \):
     - At \( x = 0 \), \( P(0) = 0^4 + 0^3 + 2(0)^2 + 0 + 2 = 2 \), which is also positive.

Given this information and the nature of the polynomial, **\( P(x) \) does not change sign in the interval \( (-1, 1) \)**. Therefore, it cannot have a root in this interval.

### Conclusion:
The statement "The equation \( P(x) = x^4 + x^3 + 2x^2 + x + 2 \) has at least one root in the interval \((-1, 1)\)" is **False**.

Would you like more details or have any questions about this solution?

### Follow-up Questions:
1. How does the Intermediate Value Theorem help in determining the existence of roots?
2. What happens to \( P(x) \) if the interval were expanded to include more negative or positive values?
3. Could a polynomial like \( P(x) \) have a complex root?
4. How does the degree of the polynomial affect the number of possible real roots?
5. What methods can be used to approximate the roots of polynomials?

### Tip:
When checking for roots in an interval, evaluating the function at critical points (e.g., boundaries and midpoints) can provide insight into whether the function changes sign, indicating a root.

Explore whether the polynomial P(x) = x^4 + x^3 + 2x^2 + x + 2 has at least one root in the interval (-1, 1). Understand the application of the Intermediate Value Theorem and polynomial analysis to solve this problem. Suitable for advanced high school students and beyond.

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