Case 1.  All 4 letters in the final word are different.

         In this case, we have only 5 distinct letters to choose from, (E, A, G, L S);

         therefore, the number of possible words to form is  5*4*3*2 = 120  in this case  (the order of letters does matter !);



Case 2.  In the final word, we have 2 identical letters E and any 2 of the remaining 4 letters.

         In this case, we can select these two remaining letters by   = 6 different ways,

         and we can arrange then 4 letters with two repeating undistinguishable Es by  

              =  = 12 different distinguishable ways.


         Combining everything altogether, we have 6*12 = 72 different words in Case 2.



Cases (1) and (2) are the disjoint sets of words; therefore, the answer to the problem's question is 120 + 72 = 192.


ANSWER.  192 different / (distinguished) words of the length 4 can be formed.