How's this?

$(\sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} (-2n+k) x^{-2 n+k-1}) - (\sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} x^{-2 (n+1)+k}) =$

$= (\sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} (-2n+k) x^{-2(n+1)+k+1}) - (\sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} x^{-2 (n+1)+k}) =$

$= (\sum _{k'=1}^{n} (k'-1)! \binom{n}{k'-1} \binom{n-1}{k'-1} (-2n+k'-1) x^{-2(n+1)+k'}) - (\sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} x^{-2 (n+1)+k}) =$

$= (\sum _{k'=0}^{n} (k'-1)! \binom{n}{k'-1} \binom{n-1}{k'-1} (-2n+k'-1) x^{-2(n+1)+k'}) - (\sum _{k=0}^{n} k! \binom{n}{k} \binom{n-1}{k} x^{-2 (n+1)+k}) =$

$= \sum _{k=0}^{n} ((k-1)! \binom{n}{k-1} \binom{n-1}{k-1} (-2n+k-1) - k! \binom{n}{k} \binom{n-1}{k}) x^{-2 (n+1)+k}$

Then $(k-1)! \binom{n}{k-1} \binom{n-1}{k-1} (-2n+k-1) - k! \binom{n}{k} \binom{n-1}{k} =$

$= \frac{(k-1)!n!(n-1)!(-2n+k-1)}{(n-k+1)!(k-1)!(n-k)!(k-1)!} - \frac{k!n!(n-1)!}{(n-k)!k!k!(n-k-1)!} =$

$= \frac{n!(n-1)!(-2n+k-1)k}{(n-k+1)!(n-k)!k!} - \frac{n!(n-1)!(n-k)(n-k+1)}{(n-k)!k!(n-k+1)!} =$

$= \frac{n!(n-1)!}{(n-k+1)!(n-k)!k!} ((-2n+k-1)k - (n-k)(n-k+1)) =$

$= \frac{-n(n+1)n!(n-1)!}{(n-k+1)!(n-k)!k!} =$

$= -\frac{(n+1)!}{(n-k+1)!} \binom{n}{k} =$

$= -k! \binom{n+1}{k} \binom{n}{k}$

How's this?

$$\left(\sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} \left(-2n+k\right) x^{-2 n+k-1}\right) - \left(\sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} x^{-2 \left(n+1\right)+k}\right) =$$

$$= \left(\sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} \left(-2n+k\right) x^{-2\left(n+1\right)+k+1}\right) - \left(\sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} x^{-2 \left(n+1\right)+k}\right) =$$

$$= \left(\sum _{k'=1}^{n} \left(k'-1\right)! \binom{n}{k'-1} \binom{n-1}{k'-1} \left(-2n+k'-1\right) x^{-2\left(n+1\right)+k'}\right) - \left(\sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} x^{-2 \left(n+1\right)+k}\right) =$$

$$= \left(\sum _{k'=0}^{n} \left(k'-1\right)! \binom{n}{k'-1} \binom{n-1}{k'-1} \left(-2n+k'-1\right) x^{-2\left(n+1\right)+k'}\right) - \left(\sum _{k=0}^{n} k! \binom{n}{k} \binom{n-1}{k} x^{-2 \left(n+1\right)+k}\right) =$$

$$= \sum _{k=0}^{n} \left(\left(k-1\right)! \binom{n}{k-1} \binom{n-1}{k-1} \left(-2n+k-1\right) - k! \binom{n}{k} \binom{n-1}{k}\right) x^{-2 \left(n+1\right)+k}$$

Then $$\left(k-1\right)! \binom{n}{k-1} \binom{n-1}{k-1} \left(-2n+k-1\right) - k! \binom{n}{k} \binom{n-1}{k} =$$

$$= \frac{\left(k-1\right)!n!\left(n-1\right)!\left(-2n+k-1\right)}{\left(n-k+1\right)!\left(k-1\right)!\left(n-k\right)!\left(k-1\right)!} - \frac{k!n!\left(n-1\right)!}{\left(n-k\right)!k!k!\left(n-k-1\right)!} =$$

$$= \frac{n!\left(n-1\right)!\left(-2n+k-1\right)k}{\left(n-k+1\right)!\left(n-k\right)!k!} - \frac{n!\left(n-1\right)!\left(n-k\right)\left(n-k+1\right)}{\left(n-k\right)!k!\left(n-k+1\right)!} =$$

$$= \frac{n!\left(n-1\right)!}{\left(n-k+1\right)!\left(n-k\right)!k!} \left(\left(-2n+k-1\right)k - \left(n-k\right)\left(n-k+1\right)\right) =$$

$$= \frac{-n\left(n+1\right)n!\left(n-1\right)!}{\left(n-k+1\right)!\left(n-k\right)!k!} =$$

$$= -\frac{\left(n+1\right)!}{\left(n-k+1\right)!} \binom{n}{k} =$$

$$= -k! \binom{n+1}{k} \binom{n}{k}$$