The problem asks to factor the linear expression $4x + 12$. We need to find the correct factored form among the given options.

1. Problem Description

The problem asks to factor the linear expression $4x + 12$. We need to find the correct factored form among the given options.

2. Solution Steps

We need to find the greatest common factor (GCF) of the terms $4x$ and $12$.

The factors of $4x$ are $1, 2, 4, x, 2x, 4x$.

The factors of $12$ are $1, 2, 3, 4, 6, 12$.

The GCF of $4x$ and $12$ is $4$.

Now, we factor out the GCF from the expression $4x + 12$:

$4x + 12 = 4(x) + 4(3) = 4(x+3)$

Let us examine the options given:

$4(x+3)$ - This is the correct factorization.

$2(2x+6)$ - Expanding this gives $4x + 12$, so this is also correct. However, this is not fully factored since $2x+6$ has a common factor of

2. $3(x+4)$ - Expanding this gives $3x + 12$, which is not equal to $4x+12$.

$4x(1+3)$ - Simplifying this gives $4x(4) = 16x$, which is not equal to $4x + 12$.

Since the problem asks to factor the expression, we are looking for the form where the GCF has been factored out. The expression $4(x+3)$ is the most simplified fully factored form of the expression. The other equivalent expression is $2(2x+6)$ but the content inside the parenthesis is not fully factored, hence $4(x+3)$ is the better answer among the available options.

3. Final Answer

$4(x+3)$