We have that $(x)_k - (x-1)_k = k (x-1)_k$. So applying the linear operator $f \mapsto xf(x) - xf(x-1)$, to the identity $$ \sum_{k=1}^{k=n} \genfrac\{\}{0pt}{}{n}{k}(x)_k = x^n $$ we get that $$\sum_{k = 1}^n \genfrac\{\}{0pt}{}{n}{k} k (x)_k = x^{n+1} - x(x-1)^n.$$
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