Question

How do you factor `9a^2 + 12ab - 5b^2`?

Answer 1

`9a^2+12ab-5b^2 = (3a+5b)(3a-b)`

You can tell from the negative sign on the `b^2` term that the signs in the linear factors must be different from one another - one `+` and one `-`. Also `5` only has divisors `+-1` and `+-5` so the factors would either be of the form `(ma+5b)(na-b)` or `(ma-5b)(na+b)` for some `m` and `n` such that `mn = 9`. I suspected that `m = n = 3` so tried those possibilities first, quickly finding the factorisation.