Question

Polynomial division

Answer 1

This is a great place and time to apply synthetic division.  Since you are dividing the 3rd order poly by (a+2), let the divisor in synth. div. be -2.

Then:
      _________________
 -2 /   1   6   12    8

     _____-2_-8    -8____________
          1   4     4     0

Since there is no remainder, -2 is a root and (x+2) is a factor of the poly.

The quotient comes from  those coefficinets 1  4  4  

and is    1a^2 + 4a + 4, whose factors are (a+2)^2.