# Angular momentum of a system about the center of mass

Note that $$\boldsymbol p'_i:=\boldsymbol p_i-m_i\dot{\boldsymbol R}$$. Therefore, $$\sum_i \boldsymbol{r}_i^\prime \times \boldsymbol{p}_i-\sum_i \boldsymbol{r}_i^\prime \times \boldsymbol{p}_i^\prime\equiv\sum_im_i\boldsymbol r'_i\times\dot{\boldsymbol R}$$ which is easily seen to vanish, inasmuch as $$\sum_im_i\boldsymbol r'_i\equiv0$$ by definition.

Thus, the expression in the OP and the one in Goldstein are identical.