$2.348 A=\left|\frac{|x-2|+4}{x-2}\right| \cdot\left(x^{2}-4\right)$.
Solution. $A=\frac{(|x-2|+4)}{|x-2|} \cdot(x-2)(x+2)$.

Consider 2 cases.

.1) $\left\{\begin{array}{l}x>2, \\ A=\frac{(x-2+4)}{x-2} \cdot(x+2)(x-2)\end{array} \Leftrightarrow\left\{\begin{array}{l}x>2, \\ A=(x+2)^{2} .\end{array}\right.\right.$

Answer: if $x>2$, then $A=(x+2)^{2}$.