The question in the image appears to be a math problem involving a quadratic equation and a condition for finding the value of $m$ that makes a specific expression minimal. Here's a breakdown of what the problem is asking:

The quadratic equation is:

$2x^2 + 2mx + m^2 - 2 = 0$

This equation has two roots, denoted $x_1$ and $x_2$. The goal is to find the value of $m$ such that the expression $A = |x_1| + |x_2| - 4$ is minimized.

To solve this, we'll follow these steps:

1. Use the quadratic formula to find $x_1$ and $x_2$.
2. Determine the condition for the minimum value of $A$ based on these roots.

I’ll start by solving the quadratic equation and then proceed with minimizing the given expression. Would you like me to walk through this step by step or provide the final answer directly?