Since in general:
#color(white)("XXX")x^2+(a+b)x+ab=(x+a)(x+b)#
Given
#color(white)("XXX")x^2+10x-56#
we are looking for numbers #a# and #b# such that
[1]#color(white)("XXX")a+b=10#
and
[2]#color(white)("XXX")ab=-54#
Since #ab# is negative
#color(white)("XXX")#one of #a# and #b# must be negative and the other positive.
Since #a+b# is positive
#color(white)("XXX")#the larger of #abs(a)# and #abs(b)# must be positive.
So we need to look for 2 factors of #54# such that
#color(white)("XXX")#the difference in their absolute values is #10#
#{:
(color(blue)("Factors of 54"),color(white)("XX"),color(blue)("Difference between factors"),),
(2xx27,,color(white)("XX")25,),
(color(red)(4xx14),,color(white)("XX")10,color(red)("This is what we need")),
(8xx7,,color(white)("XX")1,),
(3xx18,,color(white)("XX")15,),
(6xx9,,color(white)("XX")3,)
:}#
So #abs(a)# and #abs(b)# are #4# and #14#
The larger, #14#, is positive and the smaller,#4#, is negative
