The first part of this problem is an application of the Factor Theorem.
Factor Theorem: (x - a) is a factor of the polynomial p(x) if and only if
p(a) = 0.
Hence to show that (x+2) is a factor of your polynomial f(x) you only need verify that f(-2) = 0.
For the second question I would divide f(x) by (x + 2) to get
f(x) = (x + 2) g(x)
and then look at the quadratic g(x).