To factor the quadratic expression \(2x^2 + 7x + 6\), we follow these steps:

1. **Identify the terms**:
   - The coefficient of \(x^2\) (leading term) is \(2\).
   - The coefficient of \(x\) (middle term) is \(7\).
   - The constant term is \(6\).

2. **Multiply the leading coefficient and the constant**:
   \[
   2 \times 6 = 12
   \]
   
3. **Find two numbers that multiply to \(12\) and add up to \(7\)**:
   - The pair of numbers is \(3\) and \(4\), because:
     \[
     3 \times 4 = 12 \quad \text{and} \quad 3 + 4 = 7
     \]

4. **Rewrite the middle term using these numbers**:
   \[
   2x^2 + 3x + 4x + 6
   \]

5. **Factor by grouping**:
   - Group the first two terms and the last two terms:
     \[
     (2x^2 + 3x) + (4x + 6)
     \]
   - Factor out the greatest common factor from each group:
     \[
     x(2x + 3) + 2(2x + 3)
     \]

6. **Factor out the common binomial factor**:
   \[
   (x + 2)(2x + 3)
   \]

So, the fully factored form of \(2x^2 + 7x + 6\) is:
\[
(x + 2)(2x + 3)
\]

Would you like any further details or have any questions?

Here are 5 related questions:
1. How do you factor a quadratic with a negative middle term?
2. How can you check if a factored expression is correct?
3. What are the methods for solving a quadratic equation after factoring?
4. Can all quadratics be factored using integers?
5. How would the process differ if the quadratic had a leading coefficient of 1?

**Tip**: Always double-check your factors by expanding them to verify if they match the original expression.

Learn how to fully factor the quadratic expression 2x^2 + 7x + 6 with a step-by-step explanation. Understand key algebraic concepts like factoring by grouping and apply the quadratic equation formula. Ideal for students in Grades 8-10.

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