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Let's factor each of these quadratic expressions one by one.

### Given Quadratic Expressions
1. **\( 2x^2 + 13x + 15 \):**
   - We need to find two numbers that multiply to \( 2 \times 15 = 30 \) and add up to \( 13 \). These numbers are \( 10 \) and \( 3 \).
   - Rewrite the middle term using these numbers: \( 2x^2 + 10x + 3x + 15 \).
   - Group the terms: \( (2x^2 + 10x) + (3x + 15) \).
   - Factor each group: \( 2x(x + 5) + 3(x + 5) \).
   - Combine: \( (2x + 3)(x + 5) \).

2. **\( 2x^2 - 11x + 12 \):**
   - Find two numbers that multiply to \( 2 \times 12 = 24 \) and add up to \( -11 \). These numbers are \( -8 \) and \( -3 \).
   - Rewrite the middle term: \( 2x^2 - 8x - 3x + 12 \).
   - Group the terms: \( (2x^2 - 8x) + (-3x + 12) \).
   - Factor each group: \( 2x(x - 4) - 3(x - 4) \).
   - Combine: \( (2x - 3)(x - 4) \).

3. **\( 2x^2 - 11x + 15 \):**
   - Find two numbers that multiply to \( 2 \times 15 = 30 \) and add up to \( -11 \). These numbers are \( -6 \) and \( -5 \).
   - Rewrite the middle term: \( 2x^2 - 6x - 5x + 15 \).
   - Group the terms: \( (2x^2 - 6x) + (-5x + 15) \).
   - Factor each group: \( 2x(x - 3) - 5(x - 3) \).
   - Combine: \( (2x - 5)(x - 3) \).

4. **\( 2x^2 + 15x + 7 \):**
   - Find two numbers that multiply to \( 2 \times 7 = 14 \) and add up to \( 15 \). These numbers are \( 14 \) and \( 1 \).
   - Rewrite the middle term: \( 2x^2 + 14x + x + 7 \).
   - Group the terms: \( (2x^2 + 14x) + (x + 7) \).
   - Factor each group: \( 2x(x + 7) + 1(x + 7) \).
   - Combine: \( (2x + 1)(x + 7) \).

5. **\( 2x^2 + 15x + 18 \):**
   - Find two numbers that multiply to \( 2 \times 18 = 36 \) and add up to \( 15 \). These numbers are \( 12 \) and \( 3 \).
   - Rewrite the middle term: \( 2x^2 + 12x + 3x + 18 \).
   - Group the terms: \( (2x^2 + 12x) + (3x + 18) \).
   - Factor each group: \( 2x(x + 6) + 3(x + 6) \).
   - Combine: \( (2x + 3)(x + 6) \).

6. **\( 2x^2 + 9x - 18 \):**
   - Find two numbers that multiply to \( 2 \times (-18) = -36 \) and add up to \( 9 \). These numbers are \( 12 \) and \( -3 \).
   - Rewrite the middle term: \( 2x^2 + 12x - 3x - 18 \).
   - Group the terms: \( (2x^2 + 12x) + (-3x - 18) \).
   - Factor each group: \( 2x(x + 6) - 3(x + 6) \).
   - Combine: \( (2x - 3)(x + 6) \).

### Final Factored Forms
- \( 2x^2 + 13x + 15 = (2x + 3)(x + 5) \)
- \( 2x^2 - 11x + 12 = (2x - 3)(x - 4) \)
- \( 2x^2 - 11x + 15 = (2x - 5)(x - 3) \)
- \( 2x^2 + 15x + 7 = (2x + 1)(x + 7) \)
- \( 2x^2 + 15x + 18 = (2x + 3)(x + 6) \)
- \( 2x^2 + 9x - 18 = (2x - 3)(x + 6) \)

Would you like more details on any of these steps or have any questions?

### Related Questions
1. How do you factor trinomials where the leading coefficient is greater than 1?
2. What is the difference between factoring a quadratic trinomial and solving it?
3. How do you determine if a quadratic expression can be factored over the integers?
4. What are some special cases in factoring quadratic expressions (e.g., perfect square trinomials)?
5. Can all quadratic expressions be factored into linear factors?

### Tip
When factoring quadratics, always check if you can factor out a greatest common factor (GCF) first before proceeding with other methods.

Learn how to factor quadratic trinomials such as 2x^2 + 13x + 15, 2x^2 - 11x + 12, and others with step-by-step instructions. This tutorial covers key algebraic concepts including the AC method and factoring techniques for Grades 8-10.

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