$\displaystyle P_n^r+rP_n^{r-1}=\frac{n!}{(n-r)!}+r\cdot\frac{n!}{(n-r+1)!}=$
$\displaystyle =\frac{n!(n-r+1)}{(n-r+1)!}+\frac{n!r}{(n-r+1)!}=\frac{n!(n-r+1+r)}{(n-r+1)!}=\frac{(n+1)!}{(n-r+1)!}=P_{n+1}^r$
Now we use this formula to prove the second identity.
$\displaystyle P_n^r=P_{n-1}^r+rP_{n-1}^{r-1}=P_{n-2}^r+rP_{n-2}^{r-1}+r(P_{n-2}^{r-1}+(r-1)P_{n-2}^{r-2})=$
$\displaystyle =P_{n-2}^r+2rP_{n-2}^{r-1}+r(r-1)P_{n-2}^{r-2}$