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The question asks us to determine which of the three given expressions can be an integer, based on the fact that \( \sqrt{5} + x \) is an integer for some value of \( x \).

Let's analyze each statement one by one:

### Given: 
\( \sqrt{5} + x \) is an integer.

So, \( x \) must be of the form:
\[
x = -\sqrt{5} + n
\]
where \( n \) is an integer (because \( \sqrt{5} + x \) is an integer only if \( x \) cancels out the irrational part \( \sqrt{5} \)).

Now let's check each option:

### 1. \( \sqrt{5} - x \)
We know that \( x = -\sqrt{5} + n \), so:
\[
\sqrt{5} - x = \sqrt{5} - (-\sqrt{5} + n) = \sqrt{5} + \sqrt{5} - n = 2\sqrt{5} - n
\]
This expression involves \( 2\sqrt{5} \), which is irrational. Therefore, this expression cannot be an integer.

### 2. \( \sqrt{5} \cdot x + 3\sqrt{5} \)
Substitute \( x = -\sqrt{5} + n \):
\[
\sqrt{5} \cdot x + 3\sqrt{5} = \sqrt{5}(-\sqrt{5} + n) + 3\sqrt{5} = -5 + n\sqrt{5} + 3\sqrt{5}
\]
\[
= -5 + (n + 3)\sqrt{5}
\]
This expression contains the term \( (n + 3)\sqrt{5} \), which is irrational unless \( n = -3 \). In that case, the expression simplifies to:
\[
-5 + (-3 + 3)\sqrt{5} = -5
\]
which is an integer.

### 3. \( x^2 + 4\sqrt{5} \)
Substitute \( x = -\sqrt{5} + n \):
\[
x^2 + 4\sqrt{5} = (-\sqrt{5} + n)^2 + 4\sqrt{5} = (5 - 2n\sqrt{5} + n^2) + 4\sqrt{5}
\]
\[
= 5 + n^2 + (4 - 2n)\sqrt{5}
\]
This expression contains \( (4 - 2n)\sqrt{5} \), which is irrational unless \( n = 2 \). If \( n = 2 \), the expression becomes:
\[
5 + 2^2 + (4 - 2(2))\sqrt{5} = 5 + 4 + 0 = 9
\]
which is an integer.

### Conclusion:
The expressions that can be integers are options II and III. Therefore, the correct answer is **E) II ve III**.

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Do you want more details or have any questions?

Here are 5 related questions you might want to explore:
1. How can we prove whether an expression is rational or irrational?
2. What are other examples of expressions involving irrational numbers that simplify to integers?
3. What is the importance of the property of square roots of non-perfect squares being irrational?
4. How do we solve problems involving mixed rational and irrational terms?
5. How can we represent irrational numbers in terms of continued fractions?

**Tip:** When dealing with square roots of non-perfect squares, always remember they are irrational unless combined in specific ways that cancel out the irrational part.

√5 + x ifadesinin değeri bir tam sayıya eşit olduğuna göre, aşağıdaki ifadelerden hangileri tam sayı olabilir?

Analyze and solve the algebraic problem where √5 + x is an integer. Explore how to determine which expressions involving square roots and algebraic manipulations result in integer values.

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