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Let's go ahead and factor these quadratics completely.

### e. \( 2x^2 + 7x + 6 \)

We are looking for two numbers that multiply to \( 2 \times 6 = 12 \) and add up to \( 7 \). The numbers are \( 3 \) and \( 4 \).

So, factor:

\[
2x^2 + 7x + 6 = 2x^2 + 3x + 4x + 6
\]
\[
= x(2x + 3) + 2(2x + 3) = (2x + 3)(x + 2)
\]

**Factored form: \( (2x + 3)(x + 2) \)**

---

### f. \( 2x^2 - 7x + 6 \)

We need two numbers that multiply to \( 2 \times 6 = 12 \) and add up to \( -7 \). These numbers are \( -3 \) and \( -4 \).

So, factor:

\[
2x^2 - 7x + 6 = 2x^2 - 3x - 4x + 6
\]
\[
= x(2x - 3) - 2(2x - 3) = (2x - 3)(x - 2)
\]

**Factored form: \( (2x - 3)(x - 2) \)**

---

### g. \( 2x^2 + 5x + 2 \)

We are looking for two numbers that multiply to \( 2 \times 2 = 4 \) and add up to \( 5 \). The numbers are \( 1 \) and \( 4 \).

So, factor:

\[
2x^2 + 5x + 2 = 2x^2 + 4x + x + 2
\]
\[
= 2x(x + 2) + 1(x + 2) = (2x + 1)(x + 2)
\]

**Factored form: \( (2x + 1)(x + 2) \)**

---

### h. \( 2x^2 + 11x + 12 \)

We need two numbers that multiply to \( 2 \times 12 = 24 \) and add up to \( 11 \). These numbers are \( 3 \) and \( 8 \).

So, factor:

\[
2x^2 + 11x + 12 = 2x^2 + 3x + 8x + 12
\]
\[
= x(2x + 3) + 4(2x + 3) = (2x + 3)(x + 4)
\]

**Factored form: \( (2x + 3)(x + 4) \)**

---

### i. \( 2x^2 + 13x + 15 \)

We need two numbers that multiply to \( 2 \times 15 = 30 \) and add up to \( 13 \). The numbers are \( 3 \) and \( 10 \).

So, factor:

\[
2x^2 + 13x + 15 = 2x^2 + 3x + 10x + 15
\]
\[
= x(2x + 3) + 5(2x + 3) = (2x + 3)(x + 5)
\]

**Factored form: \( (2x + 3)(x + 5) \)**

---

Let me know if you'd like me to continue with the remaining expressions, or if you have any questions about the ones I solved! Here are five questions for deeper understanding:

1. What steps are involved in factoring a quadratic expression?
2. How do you find the two numbers needed for splitting the middle term?
3. Why is it necessary to factor completely when solving quadratic equations?
4. How can factoring be used to solve quadratic equations?
5. What happens when a quadratic expression cannot be factored over the integers?

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

Learn how to factor quadratic equations such as 2x^2 + 7x + 6 and 2x^2 - 7x + 6 through a detailed step-by-step process. Suitable for students in Grades 9-11, this guide will help solidify factoring concepts and algebraic techniques.

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