The particular solution of the recurrence relation I cannot find out why the particular solution of $a_n=2a_{n-1} +3n$ is $a_{n}=-3n-6$ 
here is the how I solve the relation
$a_n-2a_{n-1}=3n$
as $\beta (n)= 3n$ 
using direct guessing 
$a_n=B_1 n+ B_2$
$B_1 n+ B_2 - 2 (B_1 n+ B_2) = 3n$
So $B_1 = -3$, $B_2 = 0$
the particular solution is $a_n = -3 n$
and the homo. solution is $a_n = A_1 (-2)^n$
Why it is wrong??

using direct guessing 
$a_n=B_1 n+ B_2$
$B_1 n+ B_2 - 2 (B_1 (n-1)+ B_2) = 3n$
then
$B_1 - 2B_1 = 3$
$2 B_1 - B_2 =0$
The solution will be
$B_1 = -3, B_2=-6$
