bh[y_,h_,n_]:=Product[y-k h,{k,0,n-1}]

rhs[x_,b_,nn_]:=Sum[Binomial[nn,j]bh[b,h,nn-j]bh[x,h,j],{j,0,nn}]

rhs[x,b,4]//Expand

bh[x+b,h,4]//Expand