The number of diagonals for any n-sided convex polygon is  .


So, if "m" and "n" are the numbers of sides in our polygons, then the problem gives you two equations 


    n - m = 4                     (1)

     -  = 30       (2)


Step 1.  Multiply equation (2) by 2 (both sides).  You will get


    n*(n-3) - m*(m-3) = 60.      (3)


Step 2.  From equation (1) express n = m + 4 and substitute it into equation (3), replacing and excluding m.  You will get


    (m+4)*(m+1) - m*(m-3) = 60.


Simplify it


    m^2 + 4m + m + 4 - m^2 + 3m = 60


    8m                          = 56


     m                          = 56/8 = 7.


ANSWER.  The polygons have 7 and 11 sides.


CHECK.    -  =  -  = 11*4 - 7*2 = 44 - 14 = 30.    ! Correct !