here's one without induction, bit sketchy though:
[(x+2)(x+4)...(x+2n) + (-1^n)(x-2)(x-4)...(x-2n)]/2
=sum of the terms involving odd powers of x in the expansion of
[(x+2)(x+4)...(x+2n)]
this can be proved by expansion (remember the relation between sum of zeroes and coefficients)
if...