If y = 2 - x1 + x, then what is the value of 1y + 1 + 2y + 1y^2 - 1?

Answer: Option C

$$\begin{array}{rl}& \frac{1}{y+1}+\frac{2y+1}{y^2-1}\\ & =\frac{y-1+2y+1}{y^2-1}\\ & =\frac{3y}{y^2-1}\\ & =\frac{3\left(\frac{2-x}{1+x}\right)}{{\left(\frac{2-x}{1+x}\right)}^2-1}\\ & =\frac{3\left(2-x\right)\times \left(1+x\right)}{4+x^2-4x-1-x^2-2x}\\ & =\frac{3\left(2-x\right)\times \left(1+x\right)}{-6x+3}\\ & =\frac{3\left(2-x\right)\times \left(1+x\right)}{3\left(1-2x\right)}\\ & =\frac{\left(2-x\right)\left(1+x\right)}{\left(1-2x\right)}\\ & \\ & \mathbf{Alternate}\mathbf{solution}:\\ & y=\frac{2-x}{1+x}\\ & \text{Put }x=0,\\ & y=2\\ & \frac{1}{y+1}+\frac{2y+1}{y^2-1}\\ & =\frac{1}{3}+\frac{5}{3}\\ & =\frac{6}{3}\\ & =2\\ & \text{Go through option}\\ & \frac{\left(1+x\right)\left(2-x\right)}{\left(1-2x\right)}\\ & =\frac{1\times 2}{1}\\ & =2\end{array}$$