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The image contains a set of mathematical factoring problems. Let's break them down and solve them step by step:

### 10. Factorize the polynomial \( P(x, y) = x^4 - x^3y - 7x^2y^2 - xy^3 + 6y^4 \)
We are asked to find the factorization and the sum of coefficients of a prime factor.

1. We will attempt to factor the polynomial. A common approach is to group terms or check for patterns like quadratic forms, but in this case, we will directly factor it.

Factorization of the given polynomial:
\[
P(x, y) = (x^2 - xy - 3y^2)(x^2 + xy - 2y^2)
\]

2. Now, we calculate the sum of the coefficients of each factor.

- For \( x^2 - xy - 3y^2 \), the sum of coefficients is: \( 1 + (-1) + (-3) = -3 \).
- For \( x^2 + xy - 2y^2 \), the sum of coefficients is: \( 1 + 1 + (-2) = 0 \).

3. The smallest sum of coefficients among the prime factors is \( -3 \), so the correct answer is \( -3 \).

### 11. Factorize \( P(x) = x^4 + x^2 + 1 \)
We need to find the number of prime factors for this polynomial.

1. First, recognize that \( x^4 + x^2 + 1 \) is a common quadratic form in disguise. We can rewrite it as:
\[
P(x) = (x^2 + x + 1)(x^2 - x + 1)
\]
This is the factorization into irreducible polynomials over the real numbers.

2. The two factors \( (x^2 + x + 1) \) and \( (x^2 - x + 1) \) are prime, and thus, the total number of prime factors is **2**.

### 12. Factorize \( P(x) = x^4 + 2x^2 + 9 \)
We need to indicate a term from a prime factor.

1. We will attempt to factor this polynomial using a substitution method. Let \( z = x^2 \), so the polynomial becomes:
\[
z^2 + 2z + 9
\]
We check for factorizations but find that the quadratic does not factor nicely over the reals. Therefore, the polynomial is irreducible.

2. Since this polynomial does not factor further, we indicate that **the entire polynomial is a prime factor**.

Would you like more details on any of these steps, or do you have further questions? Here are five related questions to deepen your understanding:

1. How do we determine if a polynomial is irreducible?
2. Can you explain how to factorize higher-degree polynomials in more detail?
3. What are some strategies to simplify the sum of coefficients in factorizations?
4. How can substitution methods help in factoring complex polynomials?
5. How do we use the quadratic formula to check for irreducibility in quadratic polynomials?

**Tip**: When factoring, always look for common patterns like quadratic forms or symmetry to simplify your calculations.

Solve step by step: Factorize polynomials from the image provided.

Learn how to factorize complex polynomials like x^4 - x^3y - 7x^2y^2 - xy^3 + 6y^4 with detailed solutions. This guide covers polynomial factorization techniques, identifying irreducible polynomials, and calculating the sum of coefficients. Suitable for high school students in Grades 10-12.

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