Solutions to a cubic?

["Gerry"]

Hi

can someone help me to verify that the solutions of the cubic
ax^3 + (a - 1)x^2 - (b + 1)x + 1 - 2b = 0

can be defined as follows:

say b=na+m,
and y1=x2-x1,y2=x3-2x2+x1 (x1,x2,x3 are the roots of the cubic)

then the following quadratic form in y1,2
  (2n+1)a^2+(3m+1)a+1=3y1^2+3y1y2+y2^2
holds.

Is it always possible to determin the roots x1,2,3 by solving the
quadratic form if we know a,b,n,m?     
Thanks 
Gerry