Question 793182
If {{{4x^2+Ax+36}}} can be factored, to find the factors I would start by looking for pairs of factors of {{{4*36=144}}}
With {{{A>0}}}, A would be the sum of one of those factor pairs.
Here are the factor pairs and their sums:
{{{1*144=144}}} {{{1+144=145}}}
{{{2*72=144}}} {{{2+72=74}}}
{{{3*48=144}}} {{{3+48=51}}}
{{{4*36=144}}} {{{4+36=40}}}
{{{6*24=144}}} {{{6+24=30}}}
{{{8*18=144}}} {{{8+18=26}}}
{{{12*12=144}}} {{{12+12=24}}}
 
If {{{6x^2+Bx+24}}} can be factored, to find the factors I would start by looking for pairs of factors of {{{6*24=144}}}
With {{{B>0}}}, B would be the sum of one of those factor pairs, which were already listed above.
 
To get the largest possible value of A-B, I need to pair possible value of A with the smallest possible value of B.
So the largest difference would be {{{145-24=highlight(121)}}}
 
In that case the factorings would be
{{{4x^2+145x+36=(4x+1)(x+36)}}} and {{{6x^2+24x+24=6(x+2)^2}}}