On Page 782 in Abstract Algebra by Dummit Foote, they claim that we may inductively define $R$-module homomorphisms $s_n : P_n \to P'_{n+1}$ such that $f_n = d'_{n+1} s_n + s_{n-1} d_n$, but never show how this is done.
I worked out exercise 4, but this just has us show that chain homotopic maps induce the same maps on homology (which is used near the end of the proof).
How are they coming up with the maps $s_n$?
First rewrite equation (17.10) as $d'_{n+1} s_n = f_n - s_{n-1} d_n$. The right side is a map $$\underbrace{f_n - s_{n-1} d_n}_{g_n} : P_n \to P'_n $$ which we already know: $f_n$ and $d_n$ are given, and $s_{n-1}$ is given by induction.
On the left side, $d'_{n+1} : P'_{n+1} \to P'_n$ is given.
We need to find $s_n : P_n \to P'_{n+1}$ so that the equation $d'_{n+1} s_n = g_n$ holds. That is exactly what the definition of a projective module tells us we can do.