Factoring High School Level Olympiad Problem Factor $x^2 - 3xy + 2y^2 + x -8y - 6$
Attempt at a solution:
I have factored these and don't know how to continue...
$x^2-3xy +2y^2 = (x - y) (x-2y)$
$x^2 + x -6 = (x + 3) (x - 2)$
$2y^2 - 8y + 6 = 2 (y - 3)(y - 1)$ 

Since the polynomial is of degree $2$, we can use the
well established "tool-set" for the study of Quadrics
or Conic Sections.
So
$$
\eqalign{
  & Q(x,y) = x^{\,2}  - 3xy + 2y^{\,2}  + x - 8y - 6 =   \cr 
  &  = \left( {x,y,1} \right)^T \left( {\matrix{
   1 & { - 3/2} & {1/2}  \cr 
   { - 3/2} & 2 & { - 4}  \cr 
   {1/2} & { - 4} & 6  \cr 
 } } \right)\left( {\matrix{
   x  \cr 
   y  \cr 
   1  \cr 
 } } \right) \cr} 
$$
But the determinant of the matrix defining the Conic
$$
{\bf A}_{\,Q}  = \left( {\matrix{
   1 & { - 3/2} & {1/2}  \cr 
   { - 3/2} & 2 & { - 4}  \cr 
   {1/2} & { - 4} & 6  \cr 
 } } \right)
$$
is not null, which means that the conic is not degenerate
and thus $Q(x,y)$ cannot be factored, not even in the complex field.
