(* the original argument of the exponent.  The expression for G is mu - 0.5 (s^2)-r+0.5 (st^2) and given the definition of alpha in the appendix equals G = (2/(s^2))alpha *)
d2=(s/st)(((xp-G tp))/(Sqrt[2 tp]));(* d_2(x prime, tau prime) G is a constant *)
c=-(((s^2)(((xp-G tpp))^2))/(4((s^2)(tp-tpp)+(st^2)tpp))); (* c(x primt, tau prime prime) *)
v=-(1/2)(d2^2)-((((x-xp))^2)/(4(t-tp)))+c; (* everything combined *)

Expand[v]

Extract the coefficients of each power of xprime
This writes the exponent as
d (((x'))^2)+bTilde x'+cTilde

d=Coefficient[Expand[v],(xp^2)]
bTilde=Coefficient[Expand[v],xp]
cTilde=Expand[v]-bTilde xp-d (xp^2)//Simplify

Next write the exponent as 
d ((((x'))^2)+(bTilde/d)x')+cTilde

(bTilde/d)

To write this as 
d((( x' + (bTilde/(2 d))))^2)
we need to subtract
((bTilde^2)/(4d))from cTilde

g=-((bTilde^2)/(4 d))+cTilde
Simplify[g]

Lets check our algebra upto this point

d(((xp+(bTilde/(2 d))))^2)+g-v//Simplify

Lets check this result with what the book claims

a=((s^2)/2)G;
booksg=((-(s^2))/(4 (st^2)tp))(((x-2 a (tp/(s^2))))^2)-((s^2)/(4((s^2)(tp-tpp)+(st^2)tpp)))(((x-2 a (tpp/(s^2))))^2)+(1/4)((((((s^2)/((st^2)tp))(x-2a (tp/(s^2)))+((s^2)/((s^2)(tp-tpp)+(st^2)tpp))(x-2a (tpp/(s^2)))))^2)/((((s^2)/(st^2))(1/tp)+(1/(t-tp))+((s^2)/((s^2)(tp-tpp)+(st^2)tpp)))));

g-booksg//Simplify