\[1)\ \frac{(n + 2)!}{n!} = \frac{(n + 2)(n + 1)n!}{n!} =\]
\[= (n + 2)(n + 1) =\]
\[= n^{2} + 2n + n + 2 = n^{2} + 3n + 2.\]
\[2)\ \left( \frac{1}{n!} - \frac{1}{(n + 1)!} \right)n! =\]
\[= \left( \frac{1}{n!} - \frac{1}{(n + 1)n!} \right)n! = 1 - \frac{1}{n + 1} =\]
\[= \frac{(n + 1) - 1}{n + 1} = \frac{n}{n + 1}.\ \]