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)##