Question

How would I solve for the real solution of 1/(x+1)^2 = 1/x+1 +2

Answer 1

Is it \[
\Large \frac{1}{(x+1)^2} = \frac{1}{x+1} + 2
\] ?
If so, multiply throughout by \(\large (x+1)^2\) and simplify and solve the quadratic equation.

Answer 2

Another (but essentially identical) way to approach this is to let \(y=\dfrac{1}{x+1}\), then you have the quadratic
\[\begin{align*}y^2&=y+2\\
y^2-y-2&=0\\
(y-2)(y+1)&=0\\
y&=2,~-1
\end{align*}\]
Back substitute and solve for \(x\).