(х-4) (х+2) >=0

To solve the inequality \((x - 4)(x + 2) \geq 0\), we can follow these steps:

1. Find the critical points: Set the expression equal to zero to find the critical points. \[ (x - 4)(x + 2) = 0 \] This gives us:

   • \(x - 4 = 0 \Rightarrow x = 4\)
   • \(x + 2 = 0 \Rightarrow x = -2\)

2. Determine the intervals: The critical points divide the number line into intervals. The intervals to test are:

   • \((-\infty, -2)\)
   • \((-2, 4)\)
   • \((4, \infty)\)

3. Test each interval:

   • For the interval \((-\infty, -2)\), choose a test point like \(x = -3\): \[ (-3 - 4)(-3 + 2) = (-7)(-1) = 7 \geq 0 \quad \text{(true)} \]
   • For the interval \((-2, 4)\), choose a test point like \(x = 0\): \[ (0 - 4)(0 + 2) = (-4)(2) = -8 \geq 0 \quad \text{(false)} \]
   • For the interval \((4, \infty)\), choose a test point like \(x = 5\): \[ (5 - 4)(5 + 2) = (1)(7) = 7 \geq 0 \quad \text{(true)} \]

4. Include the critical points: Since the inequality is \(\geq 0\), we include the points where the expression equals zero, which are \(x = 4\) and \(x = -2\).

5. Combine the results: The solution to the inequality \((x - 4)(x + 2) \geq 0\) is: \[ (-\infty, -2] \cup [4, \infty) \]

Thus, the final solution is:

\[ x \in (-\infty, -2] \cup [4, \infty) \]