Factories:
`x ^(2) + 8x+16`

Answer

To factor the expression \( x^2 + 8x + 16 \), we can follow these steps:

### Step 1: Identify the coefficients
The given expression is \( x^2 + 8x + 16 \). Here, the coefficients are:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 8 \) (coefficient of \( x \))
- \( c = 16 \) (constant term)

### Step 2: Find two numbers that multiply to \( c \) and add to \( b \)
We need to find two numbers that multiply to \( c = 16 \) and add up to \( b = 8 \).

The pairs of factors of 16 are:
- \( 1 \times 16 \)
- \( 2 \times 8 \)
- \( 4 \times 4 \)

Among these pairs, \( 4 \) and \( 4 \) add up to \( 8 \).

### Step 3: Rewrite the middle term
We can rewrite the expression by splitting the middle term using the two numbers we found:
\[ x^2 + 4x + 4x + 16 \]

### Step 4: Group the terms
Now, we group the terms:
\[ (x^2 + 4x) + (4x + 16) \]

### Step 5: Factor by grouping
Now, we can factor out the common factors from each group:
\[ x(x + 4) + 4(x + 4) \]

### Step 6: Factor out the common binomial
Now, we can see that \( (x + 4) \) is a common factor:
\[ (x + 4)(x + 4) \]
or we can write it as:
\[ (x + 4)^2 \]

### Final Result
Thus, the factorization of \( x^2 + 8x + 16 \) is:
\[ (x + 4)^2 \]