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The question asks which of the expressions involving \(\sqrt{5}\) could result in an integer value.

We are given the following expressions:

1. \(\sqrt{5} - x\)
2. \(\sqrt{5} \cdot x + 3\sqrt{5}\)
3. \(x^2 + 4\sqrt{5}\)

### Analysis:

1. **Expression I: \(\sqrt{5} - x\)**  
   For this expression to be an integer, \(x\) must be a specific value. Since \(\sqrt{5}\) is irrational, we can make the expression an integer only if \(x = \sqrt{5}\), making \(\sqrt{5} - \sqrt{5} = 0\), which is an integer.

2. **Expression II: \(\sqrt{5} \cdot x + 3\sqrt{5}\)**  
   This can be simplified to \(\sqrt{5}(x + 3)\). For this to be an integer, \(\sqrt{5}(x + 3)\) must equal 0 (since any non-zero multiple of \(\sqrt{5}\) is irrational). Thus, \(x + 3 = 0 \Rightarrow x = -3\). Substituting \(x = -3\) into the expression gives:
   \[
   \sqrt{5} \cdot (-3) + 3\sqrt{5} = 0
   \]
   Hence, the result is an integer.

3. **Expression III: \(x^2 + 4\sqrt{5}\)**  
   The term \(4\sqrt{5}\) is irrational, and no matter what value \(x\) takes, the sum will always be irrational, because adding a rational number (\(x^2\)) to an irrational number (\(4\sqrt{5}\)) cannot result in an integer.

### Conclusion:
- **Expression I** can result in an integer.
- **Expression II** can result in an integer.
- **Expression III** cannot result in an integer.

Thus, the correct answer is **B) I ve II**.

Do you have any questions or want further clarification? Here are 5 related questions:

1. How can irrational numbers affect algebraic equations involving integers?
2. Why is \(\sqrt{5}\) classified as an irrational number?
3. What are the common methods to simplify expressions with irrational numbers?
4. Can a quadratic expression involving a square root result in an integer? 
5. What are some techniques to test if an expression can result in a rational number?

**Tip**: When simplifying expressions with irrational numbers, it's useful to look for patterns that could help eliminate the irrational part (like subtraction or zero multiplication).

Which of the following expressions involving √5 could be an integer?

Explore which algebraic expressions involving √5 can result in an integer. This math problem involves properties of irrational numbers, algebraic operations, and the simplification of square root expressions. Suitable for students in Grades 9-12.

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