How do you factor ##2x^2-x-21##?
##2x^2-x-21 = (2x-7)(x+3)##
Assuming that ##2x^2-x-21##
can be written as ##(ax+-c)(bx+-d)##
##color(white)(“XXXX”)####color(white)(“XXXX”)##with ##a, b, c, d in ZZ##
Since the sign of the last term (##-21##) is negative the interior signs of the two binomials must be opposite
##color(white)(“XXXX”)####rarr## we want ##(ax-c)(bx+d)##;
and
since the sign of the middle term (##-x##) is negative
##cb > ad##
Looking at factors of 2: ##(ab) in {(2,1)}##
and factors of 21: ##(c,d) in { (3,7), (7,3), (1,21), (21,1)}##
we find
##color(white)(“XXXX”)####(a,b) = (2,1)## and ##(c,d) = (7,3)##
give us
##color(white)(“XXXX”)####a*d – b*c = (-1)##
So our factors are
##color(white)(“XXXX”)####(2x-7)(x+3)##