To solve the inequality \( x^3 - 4x^2 - 11x + 60 > 0 \), we can follow these steps:

### 1. **Find the roots of the corresponding equation**:
   \[
   x^3 - 4x^2 - 11x + 60 = 0
   \]

   To find the roots, we can try to factorize the cubic polynomial or use methods such as the Rational Root Theorem. 

### 2. **Use the Rational Root Theorem**:
   The possible rational roots are the factors of the constant term (60) divided by the factors of the leading coefficient (1). So, the possible rational roots are:
   \[
   \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60
   \]
   
   Testing these values, we find that \( x = 3 \), \( x = -4 \), and \( x = 5 \) are roots of the equation.

### 3. **Factorize the cubic polynomial**:
   Given the roots, we can factorize the polynomial:
   \[
   x^3 - 4x^2 - 11x + 60 = (x - 3)(x + 4)(x - 5)
   \]

### 4. **Determine the sign of the expression**:
   Now we solve the inequality:
   \[
   (x - 3)(x + 4)(x - 5) > 0
   \]
   To solve this, we consider the sign changes across the roots \( x = 3 \), \( x = -4 \), and \( x = 5 \). The roots divide the real line into four intervals: \( (-\infty, -4) \), \( (-4, 3) \), \( (3, 5) \), and \( (5, \infty) \).

### 5. **Test each interval**:
   - For \( x \in (-\infty, -4) \): Choose \( x = -5 \), \((x - 3)(x + 4)(x - 5) < 0\)
   - For \( x \in (-4, 3) \): Choose \( x = 0 \), \((x - 3)(x + 4)(x - 5) > 0\)
   - For \( x \in (3, 5) \): Choose \( x = 4 \), \((x - 3)(x + 4)(x - 5) < 0\)
   - For \( x \in (5, \infty) \): Choose \( x = 6 \), \((x - 3)(x + 4)(x - 5) > 0\)

### 6. **Conclusion**:
   The solution to the inequality is the union of the intervals where the product is positive:
   \[
   (-4, 3) \cup (5, \infty)
   \]

So, the solution is:
\[
\boxed{(-4, 3) \cup (5, \infty)}
\]

Would you like further details or have any questions?

### Relative Questions:
1. How do you factor a cubic polynomial?
2. What is the Rational Root Theorem and how is it applied?
3. How do you determine the sign of a polynomial across different intervals?
4. Can you always factor a cubic polynomial by hand?
5. How do you graph a cubic function to visualize its roots?
6. What are the possible scenarios for the roots of a cubic equation?
7. How do you solve inequalities with more complex polynomials?
8. What is the significance of intervals in solving polynomial inequalities?

### Tip:
When solving polynomial inequalities, it's helpful to first find the roots and then analyze the sign of the polynomial on the intervals determined by these roots.

Learn how to solve the polynomial inequality x^3 - 4x^2 - 11x + 60 > 0 with detailed step-by-step instructions. This problem involves algebraic concepts such as polynomial inequalities and the application of the Rational Root Theorem. Suitable for students in Grades 10-12.

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