To factor the trinomial \(7x^2 + 2x - 5\), follow these steps:

### Step 1: Identify the coefficients
The trinomial is of the form \(ax^2 + bx + c\), where:
- \(a = 7\)
- \(b = 2\)
- \(c = -5\)

### Step 2: Multiply \(a\) and \(c\)
Multiply the coefficients \(a\) and \(c\):
\[
7 \times (-5) = -35
\]

### Step 3: Find two numbers that multiply to \(-35\) and add to \(b = 2\)
We need to find two numbers that multiply to \(-35\) and add to \(2\). These numbers are \(7\) and \(-5\), because:
\[
7 \times (-5) = -35
\]
\[
7 + (-5) = 2
\]

### Step 4: Split the middle term
Use these numbers to split the middle term (\(2x\)):
\[
7x^2 + 7x - 5x - 5
\]

### Step 5: Factor by grouping
Now, group the terms and factor out the common factors:
\[
7x(x + 1) - 5(x + 1)
\]

### Step 6: Factor out the common binomial factor
Notice that both groups contain the common factor \((x + 1)\):
\[
(7x - 5)(x + 1)
\]

So, the factored form of the trinomial \(7x^2 + 2x - 5\) is:
\[
(7x - 5)(x + 1)
\]

Would you like further details or have any questions?

### Related Questions
1. How do you factor trinomials where the leading coefficient is 1?
2. What is the difference between factoring and solving a quadratic equation?
3. Can you explain the process of completing the square?
4. How do you apply the quadratic formula to solve any quadratic equation?
5. What are special cases of factoring, such as the difference of squares?

### Tip:
When factoring trinomials, always check your factorization by expanding the factors to ensure you get back the original expression.

Learn how to factor the trinomial 7x^2 + 2x - 5 with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring trinomials and factoring by grouping. Suitable for students in Grades 9-12.

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