Factorise `x^(2) + a^(2) + 2a + 2x + 2ax . `

Options

A. (x + a) (x + a + 2)
B. (x + a) (x - a - 2)
C. (x - a) (x + a + 2)
D. (x - a) (x - a + 2)

Explanation

To factorize the expression \( x^2 + a^2 + 2a + 2x + 2ax \), we can follow these steps:

### Step 1: Rearrange the Terms
We start by rearranging the terms for better visibility:
\[ x^2 + 2ax + a^2 + 2x + 2a \]

### Step 2: Group the Terms
Next, we can group the terms in a way that allows us to factor them easily:
\[ (x^2 + 2ax + a^2) + (2x + 2a) \]

### Step 3: Factor the First Group
The first group \( x^2 + 2ax + a^2 \) is a perfect square trinomial. We can factor it as:
\[ (x + a)^2 \]

### Step 4: Factor the Second Group
In the second group \( 2x + 2a \), we can factor out the common factor of 2:
\[ 2(x + a) \]

### Step 5: Combine the Factors
Now we can combine the factored forms:
\[ (x + a)^2 + 2(x + a) \]
We can see that \( (x + a) \) is common in both terms. So we factor out \( (x + a) \):
\[ (x + a)((x + a) + 2) \]

### Step 6: Simplify the Expression
This simplifies to:
\[ (x + a)(x + a + 2) \]

Thus, the final factored form of the expression \( x^2 + a^2 + 2a + 2x + 2ax \) is:
\[ (x + a)(x + a + 2) \]