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To solve the given polynomial and find all its real zeros, let's first restate the problem:

The polynomial is:

\[
P(x) = x^5 - 4x^4 - x^3 + 10x^2 + 2x - 4
\]

We need to find all real zeros of the polynomial. I'll go through the following steps:

1. **Check for possible rational roots** using the Rational Root Theorem.
2. **Use synthetic division** or **polynomial long division** to reduce the degree if a root is found.
3. **Solve the remaining polynomial** after reduction, possibly using the quadratic formula for smaller polynomials.

I'll start by attempting possible rational roots and proceed from there.The real roots of the polynomial \( P(x) = x^5 - 4x^4 - x^3 + 10x^2 + 2x - 4 \) are:

\[
x = -1, 2, 2 - \sqrt{2}, 2 + \sqrt{2}
\]

These include:

- A real root at \( x = -1 \),
- A double real root at \( x = 2 \),
- Two additional real roots involving square roots: \( x = 2 - \sqrt{2} \) and \( x = 2 + \sqrt{2} \).

Do you have any questions or would you like more details on the steps used? 

Here are 5 related questions:
1. How does the Rational Root Theorem help in solving polynomials?
2. What is synthetic division, and how is it used in polynomial factoring?
3. How do square root terms affect the nature of polynomial roots?
4. What are the benefits of reducing a polynomial's degree when solving for zeros?
5. Can the quadratic formula be used for solving higher-degree polynomials?

**Tip:** When working with higher-degree polynomials, always try factoring or using synthetic division before moving to more advanced methods like the quadratic formula.

Find all the real zeros of the polynomial: P(x) = x^5 - 4x^4 - x^3 + 10x^2 + 2x - 4. Use the quadratic formula if necessary. Enter answers as a comma-separated list including repetitions.

Learn how to find all real zeros of the polynomial P(x) = x^5 - 4x^4 - x^3 + 10x^2 + 2x - 4 using the Rational Root Theorem and quadratic formula. This step-by-step solution covers key algebraic techniques, including synthetic division and factoring higher-degree polynomials. Suitable for high school and college algebra students.

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