An=Rationalize[(({{-4.2, 4, 0.6, 3.8, 1.6}, {1.6, 6, 1.2, 2.6, 5.2}, {-0.6, 6, -0.2, 3.4, 3.8}, {1.2, -17, -5.6, -9.8, -12.6}, {-4.4, 1, 3.2, 2.6, 0.2}}))];
Bn=(({{1, 1}, {2, 3}, {1, 1}, {5, 2}, {7, 1}}));
Cn=(({{1, 1, 0, 2, 1}, {2, 4, 2, 1, 1}}));
K=(({{(Subscript[k, 11]), (Subscript[k, 12])}, {(Subscript[k, 21]), (Subscript[k, 22])}}));
Ac = 
        An - Bn . K . Cn;
pcs1 = 
      Collect[FullSimplify[CharacteristicPolynomial[Ac, s]], s]

w = 1/2;
pds = 
        Chop[ExpandAll[(s + w -  w I) (s + w +  w  I) (s + 
                  w)] ];
pes = (Subscript[e, 2 ]) (s^2) + (Subscript[e, 1]) s  +  
          (Subscript[e, 0]); 
pcs2 = ExpandAll[pds pes];
sol = 
      Solve[CoefficientList[pcs1, s] == 
          CoefficientList[pcs2, s], {(Subscript[k, 11]), (Subscript[k, 12]), (Subscript[k, 21]), (Subscript[e, 0]), (Subscript[e, 1]), 
          (Subscript[e, 2])}]

pefix = -pes /. sol[[1]];
peshifted = 
      Collect[ExpandAll[pefix /. s -> s - 5  w], s]

pefix = -pes /. 
        sol[[1]];     
peshifted = 
      ExpandAll[pefix /. s -> s - 5  w];
r1 = CoefficientList[peshifted, s];
cond1 = 
        Apply[And, Map[(# > 0) &, r1]];
Reduce[cond1]

pefix = -pes /. 
        sol[[2]];    
peshifted = 
      ExpandAll[pefix /. s -> s - 5  w];
r1 = CoefficientList[peshifted, s];
cond1 = 
        Apply[And, Map[(# > 0) &, r1]];
Reduce[cond1]

peshifted