Question 1075689
No assumption of factorability for the given equation:

Roots using general solution formula,
{{{x=(k+- sqrt(k^2-4*2(k+2)))/(2*2)}}}


{{{(k+- sqrt(k^2-8k-16))/(2*2)}}}


{{{cross((k+- sqrt((k-4)^2))/(2*2))}}}
{{{cross(x=(k+- (k-4))/4)}}}



Using the given ratio for the roots 3:2 the resulting ratio equation is

-
{{{(k+sqrt(k^2-8k-16))/(k-sqrt(k^2-8k-16))=3/2}}}
and algebraic steps lead to 
{{{3k^2-25k-50=0}}}


Find discriminant, {{{25^2+4*3*50=1225=35^2}}}


General solution formula to get k
{{{k=(25+- 35)/6}}}


{{{highlight(system(k=-5/3,OR,k=10))}}}





----------Mistake early in the steps makes all of the below wrong----------------

Work through the quadratic equation and find that {{{x=(k-1+- sqrt(k^2-2k-15))/4}}}.

Setup the next equation according to the ratio of the roots.
{{{(k-1+sqrt(k^2-2k-15))/(k-1-sqrt(k^2-2k-15))=3/2}}}

Simplify that and reach the equation {{{6k^2-12k-47=0}}}.
If no mistakes were made in getting to this, then solve this quadratic equation for k.  Discriminant is 1272.

General solution formula for quadratic equation results in
{{{k=6+- sqrt(318)}}}.