(x+5)*(x-1)*(x-4) <0

To solve the inequality \((x + 5)(x - 1)(x - 4) < 0\), we first need to determine the critical points, which are the values of \(x\) that make each factor equal to zero.

1. Find the critical points:
   • \(x + 5 = 0 \Rightarrow x = -5\)
   • \(x - 1 = 0 \Rightarrow x = 1\)
   • \(x - 4 = 0 \Rightarrow x = 4\)

So, the critical points are \(x = -5\), \(x = 1\), and \(x = 4\).

2. Determine the intervals: The critical points divide the number line into four intervals:

   • \((-\infty, -5)\)
   • \((-5, 1)\)
   • \((1, 4)\)
   • \((4, +\infty)\)

3. Test each interval: Choose a test point from each interval and evaluate the expression \((x + 5)(x - 1)(x - 4)\):

   • Interval \((-\infty, -5)\): Choose \(x = -6\) \[ (-6 + 5)(-6 - 1)(-6 - 4) = (-1)(-7)(-10) = -70 \quad (\text{negative}) \]
   • Interval \((-5, 1)\): Choose \(x = 0\) \[ (0 + 5)(0 - 1)(0 - 4) = (5)(-1)(-4) = 20 \quad (\text{positive}) \]
   • Interval \((1, 4)\): Choose \(x = 2\) \[ (2 + 5)(2 - 1)(2 - 4) = (7)(1)(-2) = -14 \quad (\text{negative}) \]
   • Interval \((4, +\infty)\): Choose \(x = 5\) \[ (5 + 5)(5 - 1)(5 - 4) = (10)(4)(1) = 40 \quad (\text{positive}) \]

4. Summary of test results:

   • In \((-\infty, -5)\), the expression is negative.
   • In \((-5, 1)\), the expression is positive.
   • In \((1, 4)\), the expression is negative.
   • In \((4, +\infty)\), the expression is positive.

5. Determine where the expression is negative: From our test results, \((x + 5)(x - 1)(x - 4) < 0\) in the intervals:

   • \((-\infty, -5)\)
   • \((1, 4)\)

Thus, the solution to the inequality \((x + 5)(x - 1)(x - 4) < 0\) is:

\[ x \in (-\infty, -5) \cup (1, 4) \]