Let $M$ and $m$ be the largest and the smallest values of $x$, respectively, which satisfy $4x(x - 5) \le 375$. Find $M - m$.
1. Start with the given inequality:
   \[
   4x(x - 5) \le 375
   \]

2. Expand and rearrange the inequality to standard quadratic form:
   \[
   4x^2 - 20x \le 375
   \]
   \[
   4x^2 - 20x - 375 \le 0
   \]

3. Factor the quadratic expression:
   \[
   4x^2 - 20x - 375 = 0
   \]
   To factor this, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -20\), and \(c = -375\):
   \[
   x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 4 \cdot (-375)}}{2 \cdot 4}
   \]
   \[
   x = \frac{20 \pm \sqrt{400 + 6000}}{8}
   \]
   \[
   x = \frac{20 \pm \sqrt{6400}}{8}
   \]
   \[
   x = \frac{20 \pm 80}{8}
   \]
   \[
   x = \frac{100}{8} \quad \text{or} \quad x = \frac{-60}{8}
   \]
   \[
   x = \frac{25}{2} \quad \text{or} \quad x = -\frac{15}{2}
   \]

4. The quadratic expression factors as:
   \[
   4x^2 - 20x - 375 = 4(x - \frac{25}{2})(x + \frac{15}{2})
   \]

5. Determine the intervals where the inequality holds:
   \[
   4(x - \frac{25}{2})(x + \frac{15}{2}) \le 0
   \]
   The critical points are \(x = \frac{25}{2}\) and \(x = -\frac{15}{2}\). We test the intervals \((-\infty, -\frac{15}{2})\), \((- \frac{15}{2}, \frac{25}{2})\), and \((\frac{25}{2}, \infty)\).

6. Testing the intervals:
   - For \(x \in (-\infty, -\frac{15}{2})\), both factors \( (x - \frac{25}{2}) \) and \( (x + \frac{15}{2}) \) are negative, so the product is positive.
   - For \(x \in (-\frac{15}{2}, \frac{25}{2})\), one factor is negative and the other is positive, so the product is negative.
   - For \(x \in (\frac{25}{2}, \infty)\), both factors are positive, so the product is positive.

   Therefore, the inequality \(4(x - \frac{25}{2})(x + \frac{15}{2}) \le 0\) holds for:
   \[
   x \in \left[-\frac{15}{2}, \frac{25}{2}\right]
   \]

7. Identify the largest and smallest values of \(x\):
   \[
   M = \frac{25}{2}, \quad m = -\frac{15}{2}
   \]

8. Calculate \(M - m\):
   \[
   M - m = \frac{25}{2} - \left(-\frac{15}{2}\right) = \frac{25}{2} + \frac{15}{2} = \frac{40}{2} = 20
   \]

The final answer is \(\boxed{20}\)