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To solve the given inequality:

\[
\frac{3x^2 - 5x - 2}{25x^2 - 400} < 0
\]

we will follow these steps:

1. **Factorize the numerator and denominator.**
   - **Numerator:** \(3x^2 - 5x - 2\)
     Factorizing gives: \((3x + 1)(x - 2)\).

   - **Denominator:** \(25x^2 - 400\)
     Factor out 25: \(25(x^2 - 16)\).
     Then, factor \(x^2 - 16\) as a difference of squares: \(25(x - 4)(x + 4)\).

   The inequality now becomes:
   \[
   \frac{(3x + 1)(x - 2)}{25(x - 4)(x + 4)} < 0.
   \]

2. **Identify the critical points.**
   The critical points occur where the numerator or denominator is zero:
   - Numerator zeros: \(x = -\frac{1}{3}\) and \(x = 2\).
   - Denominator zeros: \(x = 4\) and \(x = -4\).

3. **Determine the sign of the expression in each interval.**
   The critical points divide the number line into five intervals:
   - \( (-\infty, -4) \)
   - \( (-4, -\frac{1}{3}) \)
   - \( (-\frac{1}{3}, 2) \)
   - \( (2, 4) \)
   - \( (4, \infty) \)

   We test the sign of the expression in each interval by choosing test points from each region.

4. **Analyze the sign pattern.**
   - For \(x < -4\), choose \(x = -5\): The expression is positive.
   - For \(-4 < x < -\frac{1}{3}\), choose \(x = -1\): The expression is negative.
   - For \(-\frac{1}{3} < x < 2\), choose \(x = 0\): The expression is positive.
   - For \(2 < x < 4\), choose \(x = 3\): The expression is negative.
   - For \(x > 4\), choose \(x = 5\): The expression is positive.

5. **Select the intervals where the expression is negative.**
   These intervals are:
   \[
   (-4, -\frac{1}{3}) \cup (2, 4).
   \]

6. **Write the solution in interval notation.**
   The solution is:
   \[
   (-4, -\frac{1}{3}) \cup (2, 4).
   \]

Would you like more details or have any questions? 

### Related Questions
1. How do you solve rational inequalities in general?
2. What changes if the inequality was \( \geq 0 \) instead of \( < 0 \)?
3. How do you factor a quadratic expression using the quadratic formula?
4. What is the significance of critical points in rational inequalities?
5. How can we graph rational functions to visualize the solution?

### Tip
Always remember to exclude points where the denominator is zero when solving rational inequalities, as these points make the expression undefined.

Solve the following inequality: (3x^2 - 5x - 2) / (25x^2 - 400) < 0. State your answer using interval notation. Use U for union and oo for ∞.

Learn how to solve the rational inequality (3x^2 - 5x - 2) / (25x^2 - 400) < 0. This problem involves factoring quadratic expressions and analyzing intervals. Suitable for students in Grades 9-12.

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