Factories:
`m^(4) + 2m^(2) n ^(2) +n ^(4)`

Options

A. `(m ^(2)+n)^(2)`
B. `(m ^(2)+n^(2))(m^2 - n^2)`
C. `(m ^(2)+n^(2))^(2)`
D. none of these

Answer

To factor the expression \( m^4 + 2m^2n^2 + n^4 \), we can follow these steps:

### Step 1: Identify the structure of the expression
The expression \( m^4 + 2m^2n^2 + n^4 \) resembles the expanded form of the square of a binomial. We recall the formula for the square of a binomial:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]

### Step 2: Rewrite the expression
We can see that:
- \( a = m^2 \)
- \( b = n^2 \)

Thus, we can rewrite our expression as:
\[ (m^2)^2 + 2(m^2)(n^2) + (n^2)^2 \]

### Step 3: Apply the binomial square formula
According to the binomial square formula, we can factor this expression as:
\[ (m^2 + n^2)^2 \]

### Final Answer
Therefore, the factorization of \( m^4 + 2m^2n^2 + n^4 \) is:
\[ (m^2 + n^2)^2 \]

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