\(A=\dfrac{\left(x^2y-y\right)\left(y+1\right)+x^2y^2-1}{\left(x^2+y\right)\left(y+1\right)+x^2y^2+1}\)
\(A=\dfrac{x^2y+x^2+y^2-y+x^2y^2-1}{x^2y+x^2+y^2+y+x^2y^2+1}\)
\(A=\dfrac{\left(x^2y^2+x^2y\right)-\left(y^2+y\right)+\left(x^2-1\right)}{\left(x^2y^2+x^2y\right)+\left(y^2+y\right)+\left(x^2+1\right)}\)
\(A=\dfrac{x^2y\left(y+1\right)-y\left(y+1\right)+\left(x^2-1\right)}{x^2y\left(y+1\right)+y\left(y+1\right)+\left(x^2+1\right)}\)
\(A=\dfrac{\left(x^2y-y\right)\left(y+1\right)+\left(x^2-1\right)}{\left(x^2y+y\right)\left(y+1\right)+\left(x^2+1\right)}\)
\(A=\dfrac{y\left(x^2-1\right)\left(y+1\right)+\left(x^2-1\right)}{y\left(x^2+1\right)\left(y+1\right)+\left(x^2+1\right)}\)
\(A=\dfrac{\left(x^2-1\right)\left[y\left(y+1\right)+1\right]}{\left(x^2+1\right)\left[y\left(y+1\right)+1\right]}\)
\(A=\dfrac{\left(x^2-1\right)\left(y^2+y+1\right)}{\left(x^2+1\right)\left(y^2+y+1\right)}\)
\(A=\dfrac{x^2-1}{x^2+1}\)
