Telescoping series of form $\sum (n+1)\cdots(n+k)$ Wolfram Alpha is able to telescope sums of the form $\sum (n+1)\cdots(n+k)$
e.g. $(1\cdot2\cdot3) + (2\cdot3\cdot4) + \cdots + n(n+1)(n+2)$
How does it do it? 
EDIT: We can rewrite as: $\sum {(n+k)! \over n!} = \sum k!{(n+k)!\over n!k!} = \sum k!{{n+k} \choose n}$
(Thanks Daniel Fischer)
EDIT2: We can also multiply out and split sums. So e.g.
$$\sum (n-1)n(n+1) = \sum (n^3-n) = \sum n^3 - \sum n$$
But sums of powers actually seem to be more nasty than the original question, involving Bernoulli numbers.
(Thanks Claude Leibovici)
And is there any name for this particular corner of maths? (i.e. How might I go about searching the Internet for information regarding this?)
PS please could we have a 'telescoping' tag?

Also your terms can be re written as
$$n(n+1)(n+2)=\dfrac{n(n+1)(n+2)(n+3)}{4}-\dfrac{(n-1)n(n+1)(n+2)}{4}.$$
