For all a and b, what is the sum of `(a-b) ^(2) and (a +b) ^(2)` ?

Options

A. `2a ^(2)`
B. `2a ^(2) -2b ^(2)`
C. `2sa ^(2) + 2b ^(2)`
D. `2a ^(2) + 4ab+2b ^(2)`

Explanation

To find the sum of \((a - b)^2\) and \((a + b)^2\), we can follow these steps:

### Step 1: Write down the expressions
We need to calculate the sum of \((a - b)^2\) and \((a + b)^2\).

### Step 2: Expand \((a - b)^2\)
Using the formula for the square of a binomial, we have:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]

### Step 3: Expand \((a + b)^2\)
Similarly, we can expand \((a + b)^2\):
\[ (a + b)^2 = a^2 + 2ab + b^2 \]

### Step 4: Add the two expanded expressions
Now, we add the two results together:
\[ (a - b)^2 + (a + b)^2 = (a^2 - 2ab + b^2) + (a^2 + 2ab + b^2) \]

### Step 5: Combine like terms
Combining the like terms from the expression:
\[ = a^2 + a^2 + b^2 + b^2 - 2ab + 2ab \]
\[ = 2a^2 + 2b^2 \]

### Step 6: Final result
Thus, the sum of \((a - b)^2\) and \((a + b)^2\) is:
\[ 2a^2 + 2b^2 \]

### Summary
The final answer is:
\[ 2(a^2 + b^2) \]

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