2x^2+12x+16=
=2x^2+8x+4x+16
=2x(x+4) 4(x+4)
=(2x+4)(x+4)

The given expression is 2x^2 + 12x + 16.

To factorize this expression, we need to find two binomials such that their product is equal to the given expression.

Step 1: Find the common factor
The given expression has a common factor of 2. We can factor out 2 from each term:
2(x^2 + 6x + 8)

Step 2: Factorize the remaining quadratic trinomial
Now, we need to factorize the quadratic trinomial inside the parentheses, which is x^2 + 6x + 8.

We can use the AC method to factorize this trinomial. In the AC method, we need to find two numbers whose product is equal to the product of the coefficient of the squared term and the constant term, and whose sum is equal to the coefficient of the middle term.

The coefficient of the squared term is 1, and the constant term is 8. The product of these two numbers is 1 * 8 = 8.

Now, we need to find two numbers whose product is 8 and whose sum is 6 (the coefficient of the middle term). The numbers that satisfy these conditions are 2 and 4.

So, we can rewrite the middle term of the trinomial as 2x + 4x.

Now, we can rewrite the trinomial as:
x^2 + 2x + 4x + 8

Step 3: Group the terms and factor by grouping
Now, we can group the terms:
(x^2 + 2x) + (4x + 8)

We can factor out the common factor from each group:
x(x + 2) + 4(x + 2)

Now, we can see that we have a common binomial factor of (x + 2) in both terms.

So, we can factor out (x + 2) from each term:
(x + 2)(x + 4)

Step 4: Simplify the expression
We have factored the given expression as (x + 2)(x + 4).

Therefore, the expression 2x^2 + 12x + 16 can be factorized as (x + 2)(x + 4).

The correct answer is option B, (2x + 4)(x + 4).