\(A=a+\dfrac{\left(2a+x\right)\left(2+x\right)-\left(2a-x\right)\left(2-x\right)-4a}{\left(2-x\right)\left(2+x\right)}\)
\(=a+\dfrac{4a+2ax+2x+x^2-4a+2ax+2x-x^2-4a}{\left(2-x\right)\left(2+x\right)}\)
\(=a+\dfrac{4ax+4x-4a}{\left(2-x\right)\left(2+x\right)}\)
\(=\dfrac{a\left(4-x^2\right)+4ax+4x-4a}{4-x^2}\)
\(=\dfrac{4a-ax^2+4ax+4x-4a}{4-x^2}\)
\(=\dfrac{-ax^2+4ax+4x}{4-x^2}\)
\(=\dfrac{x\left(-ax+4a+4\right)}{4-x^2}\)
\(=\left[\dfrac{a}{a+1}\cdot\left(-a\cdot\dfrac{a}{a+1}+4a+4\right)\right]:\left[4-\dfrac{a^2}{\left(a+1\right)^2}\right]\)
\(=\left(\dfrac{a}{a+1}\cdot\left(\dfrac{-a^2+4a^2+8a+4}{a+1}\right)\right):\left[\dfrac{4\left(a+1\right)^2-a^2}{\left(a+1\right)^2}\right]\)
\(=\dfrac{a\left(3a^2+8a+4\right)}{4\left(a+1\right)^2-a^2}=\dfrac{a\left(3a^2+6a+2a+4\right)}{\left(2a+2\right)^2-a^2}\)
\(=\dfrac{a\left(a+2\right)\left(3a+2\right)}{\left(a+2\right)\left(3a+2\right)}=a\)