Question

How do you factor the trinomial `x² - 6x - 7`?

Answer 1

`=>(x-7)(x+1|`

Explanation:

`x^2-6x-7`
`=>x^2-7x+x-7`
`=>x(x-7)+1(x-7|`
`=>(x-7)(x+1|`

Answer 2

`x^2-6x-7= color(green)((x+1)(x-7))`

Explanation:

`(x+a)(x+b) = x^2+(a+b)x+ab`

So if `x^2-6x-7` has factors of the form `(x+a)(x+b)` then
`color(white)("XXX")a+b=-6` and
`color(white)("XXX")ab=-7`

So we are looking for factors `a` and `b` of `(-7)` which add up to `(-6)`

If we optimistically assume that `a` and `b` are integers,
the only factors of `(-7)` are
`color(white)("XXX")(a,b)=(-1,7)` which gives a sum of `(+6)`
and
`color(white)("XXX")(a,b)=(1,-7)` which gives the sum we want: `(-6)`

So `(x+a)(x+b) = (x+1)(x-7)`