the simplified form of `( x + 3) ^(2) + ( x - 1) ^(2)` is

Options

A. `( x^(2) + 2 x + 5)`
B. ` 2 ( x^(2) + 2 x + 5)`
C. `( x^(2) - 2 x + 5)`
D. ` 2 ( x^(2) - 2 x + 5)`

Answer

To simplify the expression \((x + 3)^2 + (x - 1)^2\), we will follow these steps:

### Step 1: Expand the squares
We will use the formula for the square of a binomial, which states that \((a + b)^2 = a^2 + 2ab + b^2\) and \((a - b)^2 = a^2 - 2ab + b^2\).

1. Expand \((x + 3)^2\):
\[ (x + 3)^2 = x^2 + 2 \cdot x \cdot 3 + 3^2 = x^2 + 6x + 9 \]

2. Expand \((x - 1)^2\):
\[ (x - 1)^2 = x^2 - 2 \cdot x \cdot 1 + 1^2 = x^2 - 2x + 1 \]

### Step 2: Combine the expanded forms
Now, we add the two expanded forms together:
\[ (x + 3)^2 + (x - 1)^2 = (x^2 + 6x + 9) + (x^2 - 2x + 1) \]

### Step 3: Simplify the expression
Combine like terms:
1. Combine \(x^2\) terms:
\[ x^2 + x^2 = 2x^2 \]

2. Combine \(x\) terms:
\[ 6x - 2x = 4x \]

3. Combine constant terms:
\[ 9 + 1 = 10 \]

Putting it all together, we have:
\[ 2x^2 + 4x + 10 \]

### Step 4: Factor out the common term
We can factor out a 2 from the entire expression:
\[ 2(x^2 + 2x + 5) \]

### Final Answer
Thus, the simplified form of \((x + 3)^2 + (x - 1)^2\) is:
\[ 2(x^2 + 2x + 5) \]

---