​ Synthetic division

Again...set   2r + 1  = 0   ⇒   r =  -1/2  ....this is what we need to divide by

 [ note.....we need the  "placeholder" 0   to denote that there is no r^2 term...!!!]

-1/2  [  2      1       0       20      12    ]    

                   -1      0        0       -10

        ________________________

          2      0       0        20       2

The remainder,2,  is correct....

The apparent remaining polynomial is  2r^3  + 20

If we performed polynomial division....the first term would be r^3...as follows

               r^3

2r + 1  [ 2r^4

             -(2r^4)

            _________

So...this indicates that we need to divide each term of the apparent  residual polynomal by 2, the get the correct answer of

r^3  + 10  R  [ 2 / (2r + 1) ]