Question

How do you factor the expression `4x^2+5x-9`?

Answer 1

Notice that the sum of the coefficients is zero and hence find factorisation:

`4x^2+5x-9 = (x-1)(4x+9)`

Explanation:

Notice that the sum of the coefficients is `0`. That is: `4+5-9 = 0`.

So `x = 1` is a zero and `(x-1)` a factor:

`4x^2+5x-9 = (x-1)(4x+9)`

Answer 2

I llike the 'ac' method as it is a bulletproof method.
the product of coefficients 'a' and c is -36 so we need one +ve and one -ve.

now we systematically list the factors of 36 and we'll worry about the assignment of the +ve and -ve later.

1 x 36
2 x 18
3 x 12
4 x 9 and
6 x 6

SEARCH the listing for a 'pair' of factors with one +ve and one -ve so that we get a TOTAL of 'b' or total of 5 in this case.
This will occur with -4 and +9

so now we split up the middle term (the linear term) using these values:

`4x^2-4x+9x-9`
There are now four terms and each PAIR of terms will always have a common factor (this is why this method is 'bulletproof'.

`4x(x-1)+9(x-1)`

that's pretty cool.
Now you have a sum of two terms and (x - 1) is common to both of them and can be factored again to:

(x - 1)( 4x + 9)
done