As we have b = 00(1001), we have XXb = 10(0110) that is correct in the example solution. What is that so? Maybe you see that clearer when I am writing the values of b[t]. We have b[0]=0, b[1]=0, b[2]=1, b[3]=0, b[4]=0, b[5]=1, and so on. Hence, (XXb)[t] = (Xb)[t+1] = b[t+2], and thus, (XXb)[0]=b[2]=1, (XXb)[1]=b[3]=0, (XXb)[2]=b[4]=0, (XXb)[3]=b[5]=1, and so on.