Question

How do you factor `6x^2+x-15`?

Answer 1

`6x^2+x-15`

Base factors (without regard to signs) of
`6 = S_6 = {(1xx6), (2xx3)}`

Base factors (without regard to signs) of
`15 = S_(15) = {(1xx15),(3xx5)}`

Since in the given expression the term `15` is negative
we are looking for a pair from `S_6` and another pair from `S_(15)`
that can be multiplied as
one term from `S_6` times one term from `S_(15)`
subtracted from
the other term from `S_6` times the other term from `S_(15)`
with a result of `1` (the coefficient of the `x` term).

Only a little thinking produces the pairs:
`color(red)( "("3,2")") " and " color(blue)("("3,5")")`

and the solution quickly follows
`6x^2+x-15`

`=(color(red)(3)x+color(blue)(5))(color(red)(2)x-color(blue)(3))`