The factors of the expression
`a+b+c+2sqrt(ab)-2sqrt(bc)-2sqrt(ca)` are ___________.

Options

A. `sqrt(a)-sqrt(b)+sqrt(c),sqrt(a)-sqrt(b)+sqrt(c)`
B. `sqrt(a)+sqrt(b)-sqrt(c),sqrt(a)-sqrt(b)+sqrt(c)`
C. `sqrt(a)+sqrt(b)-sqrt(c),sqrt(a)+sqrt(b)-sqrt(c)`
D. `sqrt(a)+sqrt(b)-sqrt(c),sqrt(a)+b-c`

Answer

To factor the expression \( a + b + c + 2\sqrt{ab} - 2\sqrt{bc} - 2\sqrt{ca} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ a + b + c + 2\sqrt{ab} - 2\sqrt{bc} - 2\sqrt{ca} \]

### Step 2: Group terms involving square roots
We can rearrange the expression to group the terms involving square roots:
\[ a + b + c + 2\sqrt{ab} - (2\sqrt{bc} + 2\sqrt{ca}) \]

### Step 3: Recognize a pattern
Notice that the expression resembles the expansion of a squared binomial. We can relate it to the identity:
\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx \]
where \( x = \sqrt{a} \), \( y = \sqrt{b} \), and \( z = -\sqrt{c} \).

### Step 4: Apply the identity
Using the identity, we can express our expression as:
\[ (\sqrt{a} + \sqrt{b} - \sqrt{c})^2 \]
This means we can write:
\[ a + b + c + 2\sqrt{ab} - 2\sqrt{bc} - 2\sqrt{ca} = (\sqrt{a} + \sqrt{b} - \sqrt{c})^2 \]

### Step 5: Write the factors
Thus, the factors of the expression are:
\[ (\sqrt{a} + \sqrt{b} - \sqrt{c}) \times (\sqrt{a} + \sqrt{b} - \sqrt{c}) \]
or simply:
\[ (\sqrt{a} + \sqrt{b} - \sqrt{c})^2 \]

### Final Answer
The factors of the expression \( a + b + c + 2\sqrt{ab} - 2\sqrt{bc} - 2\sqrt{ca} \) are:
\[ (\sqrt{a} + \sqrt{b} - \sqrt{c})^2 \]

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