Question 1209650
1. **Rewrite the inequality:**
   Move all terms to one side:
   x³ - 2x² - 3x + 25x² - 17x < 0
   x³ + 23x² - 20x < 0

2. **Factor out x:**
   x(x² + 23x - 20) < 0

3. **Solve the quadratic equation:**
   Find the roots of x² + 23x - 20 = 0 using the quadratic formula:
   x = (-b ± √(b² - 4ac)) / 2a
   x = (-23 ± √(23² - 4(1)(-20))) / 2(1)
   x = (-23 ± √(529 + 80)) / 2
   x = (-23 ± √609) / 2
   The roots are approximately x ≈ 0.839 and x ≈ -23.839

4. **Determine the intervals:**
   The roots of the cubic equation are x = 0, x ≈ 0.839, and x ≈ -23.839. These roots divide the number line into four intervals:
   * x < -23.839
   * -23.839 < x < 0
   * 0 < x < 0.839
   * x > 0.839

5. **Test points in each interval:**
   * x = -24: (-24)(-24² + 23(-24) - 20) < 0 => -24(576 - 552 - 20) < 0 => -24(4) < 0. True
   * x = -1: (-1)(1 - 23 - 20) < 0 => (-1)(-42) < 0. False
   * x = 0.5: (0.5)(0.25 + 11.5 - 20) < 0 => 0.5(-8.25) < 0. True
   * x = 1: (1)(1 + 23 - 20) < 0 => 4 < 0. False

6. **Write the solution in interval notation:**
   The inequality is satisfied when x < -23.839 or 0 < x < 0.839.

Final Answer: The final answer is $\boxed{(-\infty, \frac{-23 - \sqrt{609}}{2}) \cup (0, \frac{-23 + \sqrt{609}}{2})}$