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To factor the trinomial \(2x^2 + 9x + 7\), we will use the method of splitting the middle term.

### Step-by-Step Solution:

1. **Identify the coefficients:**
   - The coefficient of \(x^2\) (the leading term) is \(a = 2\).
   - The coefficient of \(x\) (the middle term) is \(b = 9\).
   - The constant term (the last term) is \(c = 7\).

2. **Find two numbers that multiply to \(a \cdot c = 2 \cdot 7 = 14\) and add to \(b = 9\).**
   - The numbers 7 and 2 multiply to 14 and add to 9.

3. **Rewrite the middle term using these two numbers:**
   \[
   2x^2 + 9x + 7 = 2x^2 + 7x + 2x + 7
   \]

4. **Group the terms and factor each group:**
   \[
   = (2x^2 + 2x) + (7x + 7)
   \]
   Factor out the common terms in each group:
   \[
   = 2x(x + 1) + 7(x + 1)
   \]

5. **Factor by grouping:**
   Notice that \((x + 1)\) is common in both groups:
   \[
   = (2x + 7)(x + 1)
   \]

### Factored Form:
The factored form of the trinomial \(2x^2 + 9x + 7\) is:
\[
(2x + 7)(x + 1)
\]

Would you like more details or have any questions?

Here are 5 related questions you might explore next:
1. How do you factor trinomials when the leading coefficient is greater than 1?
2. What is the difference between factoring by grouping and other methods of factoring?
3. Can all quadratic trinomials be factored using integers?
4. How do you solve a quadratic equation by factoring?
5. What is the quadratic formula, and how is it used to find the roots of a quadratic equation?

**Tip:** Always check your factored form by expanding it back to the original expression to ensure accuracy.

Learn how to factor the trinomial 2x^2 + 9x + 7 using the method of grouping. Detailed step-by-step solution suitable for students in Grades 9-10.

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