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To solve the quadratic inequality \(4x^2 + 7x - 2 \geq 0\) by factoring, we need to follow these steps:

### 1. Factor the quadratic expression
We need to find two numbers that multiply to \(4 \times (-2) = -8\) and add to \(7\). The numbers \(8\) and \(-1\) fit these conditions. Thus, we rewrite the middle term:

\[
4x^2 + 8x - x - 2 \geq 0
\]

Next, we group and factor:

\[
(4x^2 + 8x) - (x + 2) \geq 0
\]

Factor each group:

\[
4x(x + 2) - 1(x + 2) \geq 0
\]

Now factor out the common factor:

\[
(4x - 1)(x + 2) \geq 0
\]

### 2. Determine the critical points
Set each factor equal to zero to find the critical points:

\[
4x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{4}
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]

These critical points divide the number line into intervals: \((- \infty, -2)\), \((-2, \frac{1}{4})\), and \((\frac{1}{4}, \infty)\).

### 3. Test the intervals
We will test the sign of the expression \((4x - 1)(x + 2)\) in each interval.

- For \(x \in (-\infty, -2)\), choose \(x = -3\):
  \[
  (4(-3) - 1)((-3) + 2) = (-13)(-1) > 0
  \]

- For \(x \in (-2, \frac{1}{4})\), choose \(x = 0\):
  \[
  (4(0) - 1)((0) + 2) = (-1)(2) < 0
  \]

- For \(x \in (\frac{1}{4}, \infty)\), choose \(x = 1\):
  \[
  (4(1) - 1)(1 + 2) = (3)(3) > 0
  \]

### 4. Write the solution in interval notation
The inequality is satisfied in the intervals where the product is non-negative, which are \((- \infty, -2]\) and \([\frac{1}{4}, \infty)\).

### 5. Set-builder notation
The solution set in set-builder notation is:

\[
\{x \mid x \leq -2 \text{ or } x \geq \frac{1}{4}\}
\]

Would you like more details or have any questions? 

Here are some related questions you might find helpful:

1. How do you determine which intervals to test for signs in a quadratic inequality?
2. Can all quadratic inequalities be solved by factoring?
3. What if the quadratic expression is not factorable? How should you proceed?
4. How do the signs of the factors influence the solution of an inequality?
5. Why is it important to include the critical points when the inequality is non-strict (e.g., \( \geq 0\))?

**Tip:** Always double-check your factorization and the intervals you test to ensure accuracy in solving inequalities.

Learn how to solve the quadratic inequality 4x^2 + 7x - 2 ≥ 0 using factoring with a detailed step-by-step solution. This problem covers quadratic inequalities and factoring techniques. Suitable for high school students in Grades 9-12.

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