Question

I know the answer is C. but I need to know how to solve it!

Answer 1

Yes you are right it's C)
Notice  that f(-2) = (-2)³ +2( - 2)² + 4(-2) + 8 =0
Then x³+2x²+4x+8 is divisible by (x+2)
Perform this division & you will get:

x³+2x²+4x+8 = (x+2)(x²+4)
 The root of (x+2) is the value that render it nil, that is x=-2
  The root of (x²+4) are imaginary (not real) because
  x²+4 =0 ==> x² = -4 & x=√-4 or x= 2i (nor real).
Hope you understood the reasoning