Factorise :
`x^(2) + 3x + 2 + ax + 2a`

Answer

To factorise the expression \( x^2 + 3x + 2 + ax + 2a \), we will follow these steps:

### Step 1: Rearrange the expression
First, let's rearrange the expression to group similar terms together:
\[ x^2 + (3 + a)x + (2 + 2a) \]

### Step 2: Identify the coefficients
Now, we identify the coefficients for the quadratic expression:
- Coefficient of \( x^2 \) is 1.
- Coefficient of \( x \) is \( 3 + a \).
- Constant term is \( 2 + 2a \).

### Step 3: Factor by grouping
We will use the method of grouping. We need to express \( 3 + a \) as a sum of two numbers that multiply to give \( 1 \times (2 + 2a) \).

Let’s find two numbers that add up to \( 3 + a \) and multiply to \( 2 + 2a \).

### Step 4: Split the middle term
We can rewrite \( 3 + a \) as \( 2 + 1 + a \):
\[ x^2 + (2 + 1 + a)x + (2 + 2a) \]
This can be grouped as:
\[ x^2 + (2 + a)x + 1x + (2 + 2a) \]

### Step 5: Group the terms
Now, we can group the terms:
\[ (x^2 + (2 + a)x) + (1x + (2 + 2a)) \]

### Step 6: Factor out common terms
From the first group \( (x^2 + (2 + a)x) \), we can factor out \( x \):
\[ x(x + (2 + a)) + 1(x + (2 + 2a)) \]

### Step 7: Factor out the common binomial
Now, we can see that \( (x + (2 + a)) \) is common:
\[ (x + (2 + a))(x + 1) \]

### Final Expression
Thus, the factorised form of the expression \( x^2 + 3x + 2 + ax + 2a \) is:
\[ (x + 2 + a)(x + 1) \]