The problem states that the point $A'(x+1, -2)$ is the image of the point $A(-4, 2)$ after a rotation of $180^\circ$ around the origin $O$. We need to find the value of $x$.

1. Problem Description

The problem states that the point $A'(x+1, -2)$ is the image of the point $A(-4, 2)$ after a rotation of $180^\circ$ around the origin $O$. We need to find the value of $x$.

2. Solution Steps

When a point $(a, b)$ is rotated $180^\circ$ about the origin, its image is $(-a, -b)$.

Therefore, if $A(-4, 2)$ is rotated $180^\circ$ about the origin, its image $A'$ will be $(4, -2)$.

We are given that $A'(x+1, -2)$. Therefore, we can equate the x-coordinates:

$x + 1 = 4$

$x = 4 - 1$

$x = 3$

3. Final Answer

The value of $x$ is 3.