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Let's solve the equations step by step:

1. **\(a^2 + 12a + 11 = (a + \_\_)(a + \_\_)\):**

   Factor \(a^2 + 12a + 11\):
   - Two numbers that multiply to 11 and add to 12 are 1 and 11.
   - So, the factored form is \( (a + 1)(a + 11) \).

   **Answer: \( (a + 1)(a + 11) \)**

2. **\(b^2 + 6b + 8 = (b + 2)(b + \_\_)\):**

   Factor \(b^2 + 6b + 8\):
   - Two numbers that multiply to 8 and add to 6 are 2 and 4.
   - So, the factored form is \( (b + 2)(b + 4) \).

   **Answer: \( (b + 2)(b + 4) \)**

3. **\(c^2 - 7c + 6 = (c - \_\_)(c - \_\_)\):**

   Factor \(c^2 - 7c + 6\):
   - Two numbers that multiply to 6 and add to -7 are -1 and -6.
   - So, the factored form is \( (c - 1)(c - 6) \).

   **Answer: \( (c - 1)(c - 6) \)**

4. **\(2x^2 - 3x - 9 = (\_\_x + \_\_)(x + 3)\):**

   Factor \(2x^2 - 3x - 9\):
   - First, identify factors of the quadratic:
     - We will split the middle term: \(2x^2 - 6x + 3x - 9\)
     - Grouping: \(2x(x + 3) + 3(x + 3)\)
     - So, the factored form is \( (2x + 3)(x + 3) \).

   **Answer: \( (2x + 3)(x + 3) \)**

5. **\(3x^2 + 5x - 2 = (\_\_x - 1)(x + 2)\):**

   Factor \(3x^2 + 5x - 2\):
   - Use trial and error to factor:
     - The correct factorization is \( (3x - 1)(x + 2) \).

   **Answer: \( (3x - 1)(x + 2) \)**

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Would you like more details on any of these steps? Here are 5 related questions to extend your practice:

1. How do you factor a quadratic expression if no two numbers seem to work?
2. What is the general process for factoring a quadratic trinomial?
3. Can all quadratic expressions be factored using integers? Why or why not?
4. What are other methods to solve quadratic equations besides factoring?
5. How do you apply the distributive property to check your factoring work?

**Tip:** Always check your factored form by multiplying the factors to verify they expand back to the original quadratic.

Factor the following algebraic expressions:
1. a^2 + 12a + 11 = (a + _)(a + _)
2. b^2 + 6b + 8 = (b + _)(b + _)
3. c^2 - 7c + 6 = (c - _)(c - _)
4. 2x^2 - 3x - 9 = (_x + _)(x + 3)
5. 3x^2 + 5x - 2 = (_x - 1)(x + 2)

Learn how to factor quadratic equations and algebraic expressions step by step. This includes factoring expressions like a^2 + 12a + 11, b^2 + 6b + 8, and others. Suitable for Grades 9-10.

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