We need to solve the inequality 

\[
\frac{(x - 19)^2 (x + 1)^3}{(x - 26)^4} > 0
\]

### Step 1: Analyze the critical points
The critical points occur where the expression equals zero or is undefined. These occur when the numerator is zero or when the denominator is zero.

- **Numerator**: \((x - 19)^2 (x + 1)^3 = 0\)
  - \(x = 19\) (with multiplicity 2)
  - \(x = -1\) (with multiplicity 3)

- **Denominator**: \((x - 26)^4 = 0\)
  - \(x = 26\) (with multiplicity 4)

### Step 2: Determine the sign of the expression in each interval
These critical points divide the real number line into the following intervals:
\[
(-\infty, -1), \, (-1, 19), \, (19, 26), \, (26, \infty)
\]

To determine the sign of the expression on each interval, consider the behavior of each factor:

1. **Interval \((-∞, -1)\)**:
   - \((x - 19)^2 > 0\) (since any squared term is positive)
   - \((x + 1)^3 < 0\) (since \(x + 1 < 0\) and cubed term retains the sign)
   - \((x - 26)^4 > 0\) (since any term to an even power is positive)

   Product: \(\text{Positive} \times \text{Negative} \times \text{Positive} = \text{Negative}\)

2. **Interval \((-1, 19)\)**:
   - \((x - 19)^2 > 0\)
   - \((x + 1)^3 > 0\)
   - \((x - 26)^4 > 0\)

   Product: \(\text{Positive} \times \text{Positive} \times \text{Positive} = \text{Positive}\)

3. **Interval \((19, 26)\)**:
   - \((x - 19)^2 > 0\)
   - \((x + 1)^3 > 0\)
   - \((x - 26)^4 > 0\)

   Product: \(\text{Positive} \times \text{Positive} \times \text{Positive} = \text{Positive}\)

4. **Interval \((26, \infty)\)**:
   - \((x - 19)^2 > 0\)
   - \((x + 1)^3 > 0\)
   - \((x - 26)^4 > 0\)

   Product: \(\text{Positive} \times \text{Positive} \times \text{Positive} = \text{Positive}\)

### Step 3: Consider the equality and undefined points
- At \(x = 19\): The expression is zero (not included in the solution).
- At \(x = -1\): The expression is zero (not included in the solution).
- At \(x = 26\): The expression is undefined (not included in the solution).

### Step 4: Combine the results
The inequality is satisfied (positive) in the intervals:
\[
(-1, 19) \cup (19, 26) \cup (26, \infty)
\]

### Final Answer:
The solution to the inequality is 

\[
\boxed{(-1, 19) \cup (19, 26) \cup (26, \infty)}
\]

Would you like details on any step? 

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Here are five related questions:

1. How would the inequality change if the exponent of \((x+1)\) was even?
2. What is the impact of having odd vs. even exponents on the intervals where the expression is positive?
3. How do we determine the behavior of the inequality at the critical points?
4. How would the solution change if the inequality were non-strict (≥ instead of >)?
5. How can this method be applied to more complex rational inequalities?

**Tip:** Always check the multiplicity of the roots, as it can affect the sign of the expression across intervals!

Learn how to solve the rational inequality ((x−19)^2(x+1)^3)/(x−26)^4 > 0 with detailed step-by-step analysis. This problem covers inequalities, rational expressions, and critical point analysis, suitable for advanced high school students or college-level mathematics.

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