Question 1164256
<br>
The rational roots theorem tells us the only possible rational roots are 1, -1, 2, and -2.<br>
Synthetic division shows us that -2 is a root:<br><pre>

  -2 | 1  3  3  2
     |   -2 -2 -2
     ------------
       1  1  1  0</pre>
ANSWER: {{{a^3+3a^2+3a+2 = (a+2)(a^2+a+1)}}}<br>
Here is a different path to the answer; this is what first caught my eye when I saw the problem.<br>
{{{a^3+3a^2+3a+2 = (a^3+3a^2+3a+1)+1 = (a+1)^3+1^3}}}<br>
Use the factorization pattern {{{x^3+y^3 = (x+y)(x^2-xy+y^2)}}}:<br>
{{{(a+1)^3+1^3 = ((a+1)+1)((a+1)^2-(a+1)+1) = (a+2)(a^2+a+1)}}}<br>