@Carl Love This problem was prosed by a user a year ago on Mapleprime: looking for a fast way to find 6 integer roots. I suggested a solution at that time. I am revisting it to see if I can speedup my original solution. I enjoyed reviewing your script. It will take me much time to understand it.
You appear to have found more than 2 possible 6 integer root solutions. I may be wrong, but I substituted the values for the 7 coefficients from Sols into the equation: abs(a*x + b) + abs(c*x + d) - t*x^2 + m*x - n and calculated the equation for the various roots (x). The results were not always zero. Please see script.
I hope I understood your question correctly about the possibilities of 8 integer roots instead of 6. It never occurred to me that there will be 8 integer roots. I modified the script called Roots_with_map.mw found in my original question. I changed the statement if nops(X) = 6 to if nops(X) = 8 and added and frac(X) = 0 and frac(X) = 0 in Procedure 2. I was not able to come up solution with 8 roots. Please show me how to get 8.
Thanks again for your help, I throughly enjoyed it.