`a ^(2) + bc + ab + ac=` ?

Options

A. `(a+ b) (a+c)`
B. `(a+b) (b +c)`
C. `(b +c) (c+a)`
D. `a(a+b+c)`

Answer

To factorize the expression \( a^2 + ab + ac + bc \), we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ a^2 + ab + ac + bc \]

### Step 2: Rearrange the terms
We can rearrange the terms to group them in a way that makes factoring easier:
\[ a^2 + ab + ac + bc = a^2 + ab + ac + bc \]

### Step 3: Group the terms
Now, we will group the terms:
\[ (a^2 + ab) + (ac + bc) \]

### Step 4: Factor out the common terms
Now we will factor out the common factors from each group:
- From the first group \( a^2 + ab \), we can factor out \( a \):
\[ a(a + b) \]
- From the second group \( ac + bc \), we can factor out \( c \):
\[ c(a + b) \]

### Step 5: Combine the factored terms
Now we can combine the two factored expressions:
\[ a(a + b) + c(a + b) \]

### Step 6: Factor out the common binomial
Now we see that \( (a + b) \) is common in both terms, so we can factor it out:
\[ (a + b)(a + c) \]

### Final Answer
Thus, the factorized form of the expression \( a^2 + ab + ac + bc \) is:
\[ (a + b)(a + c) \]

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