Question 943336
{{{3*3y^2-2*3y+2*3}}}
{{{3(3y^2-2y+2)}}}


Find sensible integer combinations to try.
(3y____1)(y_____2)
Gives a 1y and a 6y.  This will not give -2y.


(3y_____2)(y______1)
Gives 2y and 3y.  This will not give -2y.


Next thing to try is roots for factoring the quadratic factor.
{{{y=(6+- sqrt(36-4*3*2))/(2*3)}}}
{{{y=(6+- sqrt(36-24))/6}}}
the discriminant is 36-24=12, which will still give an irrational discriminant.


The original expression is not factorable the way you want it, but can only go as far as  {{{highlight(3(3y^2-2y+2))}}}.


The factoring can go further, but you would need to accept irrational roots.