\(A=\frac{x}{x+1}.\left(\frac{x^3+1}{x^2-x+1}+\frac{x+1}{x}\right)=\frac{x}{x+1}.\left(\frac{\left(x+1\right)\left(x^2-x+1\right)}{x^2-x+1}+\frac{x+1}{x}\right)=\frac{x}{x+1}.\left(x+1+\frac{x+1}{x}\right)\\ =\frac{x}{x+1}.\left(\frac{x^2+x+1}{x}\right)\)
\(=\frac{x}{x+1}.\frac{\left(x+1\right)^2}{x}=x+1\)