The question from Nielsen & Chuang's Quantum Computation and Quantum Information asks to show that the action of the bit flip channel described by the quantum operation $$\epsilon(\rho)= (1 - p) \rho + p X\rho X$$ can be given an alternative operator-sum representation as: $$\epsilon (\rho) = (1-2p)\rho + 2p P_{+} \rho P_{+} + 2p P_{-} \rho P_{-} $$
where $P_{+}$ and $P_{-}$ are projectors onto the $+1$ and $-1$ eigenstates of $X$.
It is my understanding that $p$ denotes the probability of a bit flip and $(1-p)$ denotes the probability of the state remaining unchanged.
From different sources, it seems that the first line of the proof involves rewriting $\epsilon (\rho)$ as $$\epsilon (\rho) = (1-2p) \rho + p (X\rho X + I \rho I)$$
I can take the proof from there but I don't understand how we reached this first line of the proof. Any help would be greatly appreciated.
We have $1-p=1-2p+p$, using some algebraic manipulation, we have
\begin{align} \epsilon(\rho) &= (1-p)\rho + p X \rho X\\ &=(1-2p+p)\rho + pX \rho X \\ &= ( 1-2p)\rho + p\rho + pX \rho X \\ &=(1-2p)\rho + p(\rho + X\rho X)\\ &=(1-2p)\rho + p(I\rho I+X\rho X) \end{align}
Note that $I$ is the identity here.