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We are tasked with factoring the polynomial:

\[
x^4 + 5x^3 + 7x^2 - x - 2
\]

### Step-by-step solution:

1. **Search for Rational Roots**: We can use the Rational Root Theorem to find possible rational roots. The possible rational roots are given by the factors of the constant term divided by the factors of the leading coefficient. Here, the constant term is \(-2\) and the leading coefficient is \(1\), so the possible roots are:

\[
\pm 1, \pm 2
\]

2. **Test the Possible Roots**: We use synthetic division or direct substitution to test if any of these values are roots.

Let's try \(x = 1\):

\[
1^4 + 5(1^3) + 7(1^2) - 1 - 2 = 1 + 5 + 7 - 1 - 2 = 10 \quad (\text{not a root})
\]

Next, try \(x = -1\):

\[
(-1)^4 + 5(-1)^3 + 7(-1)^2 - (-1) - 2 = 1 - 5 + 7 + 1 - 2 = 2 \quad (\text{not a root})
\]

Now, try \(x = 2\):

\[
2^4 + 5(2^3) + 7(2^2) - 2 - 2 = 16 + 40 + 28 - 2 - 2 = 80 \quad (\text{not a root})
\]

Finally, try \(x = -2\):

\[
(-2)^4 + 5(-2)^3 + 7(-2)^2 - (-2) - 2 = 16 - 40 + 28 + 2 - 2 = 4 \quad (\text{not a root})
\]

None of these simple rational roots work, which suggests that factoring this polynomial will require a different method, such as grouping or numerical techniques. Based on the complexity of the roots, it is likely that the roots might be irrational or complex.

Would you like me to proceed using a more advanced factorization technique or find numerical approximations for the roots?

---

**Questions:**

1. What is the Rational Root Theorem, and how does it help in factoring polynomials?
2. Why do we test for rational roots first when factoring a polynomial?
3. What should we do if none of the rational root tests provide a valid root?
4. How does synthetic division help in finding the factors of a polynomial?
5. Are there techniques for factoring polynomials with complex or irrational roots?

**Tip:** The Rational Root Theorem is useful, but it only helps with simple rational numbers. More advanced methods like the quadratic formula or factoring by grouping can help when rational root tests fail.

Learn how to factor the polynomial x^4 + 5x^3 + 7x^2 - x - 2 using the Rational Root Theorem and synthetic division. This step-by-step solution covers key algebraic concepts for students in Grades 10-12.

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