Square root of `x + y + z + 2sqrt(xy + yz` is

Options

A. `sqrt(x) + sqrt(y + z)`
B. `sqrt(x + y + z)`
C. `sqrt(x + y) + sqrt(z)`
D. None of these

Answer

To solve the question, we need to find the square root of the expression \( x + y + z + 2\sqrt{xy + yz} \).

### Step-by-Step Solution:

1. **Identify the expression**: We start with the expression \( x + y + z + 2\sqrt{xy + yz} \).

2. **Rearranging the expression**: Notice that we can rearrange the expression to group terms that can be factored. We can write it as:
\[ x + z + y + 2\sqrt{y(x + z)} \]

3. **Recognizing a perfect square**: The expression \( y + 2\sqrt{y(x + z)} \) can be recognized as a perfect square. We can rewrite it as:
\[ (\sqrt{y} + \sqrt{x + z})^2 \]
This is because \( (\sqrt{y})^2 + ( \sqrt{x + z})^2 + 2\sqrt{y}\sqrt{x + z} = y + (x + z) + 2\sqrt{y(x + z)} \).

4. **Taking the square root**: Now we can take the square root of the entire expression:
\[ \sqrt{x + y + z + 2\sqrt{y(x + z)}} = \sqrt{(\sqrt{y} + \sqrt{x + z})^2} \]
This simplifies to:
\[ \sqrt{y} + \sqrt{x + z} \]

5. **Final expression**: Thus, the square root of the original expression is:
\[ \sqrt{y} + \sqrt{x + z} \]

### Conclusion:
The final answer is \( \sqrt{y} + \sqrt{x + z} \).