Let the answer = A, then

(1)  6x³ + x² - 7x + 2014 = A

Get 0 on the right of the given equation:

(2)  2x² + x - 2 = 0

Multiply (2) by -3x so that it will have a 
cube term that will cancel the cube term of (1)
if we add them:

(3) -6x³ - 3x² - 6x = 0

Now add equations (1) and (3)

(1)   6x³ +  x² - 7x + 2014 = A
(3)  -6x³ - 3x² - 6x        = 0
-------------------------------
(4)        -2x² -  x + 2014 = A

Now add equations (4) and (2)

(4) -2x² - x + 2014 = A            
(2)  2x² + x -    2 = 0
-----------------------
               2012 = A

The answer is 2012.

Edwin