Question

How do you factor `4x^2-32x+60`?

Answer 1

factor out 4

`4(x^2-8x+15)`

what numbers add up to -8 and multiply to 15?
-3 and -5

`4(x-3)(x-5)`

Answer 2

`4x^2-32x +60 = (2x-6)(2x-10)`

Explanation:

We can write that trinomial as

`4x^2 - 32x + 60 = 4x^2 - 32x + 64 - 4`

Now,
`4x^2 - 32x + 64 = (color(red)(2x))^2 - 2*color(red)(2x)*color(red)8 + color(red)8^2`

Knowing that
`(a-b)^2 = a^2-2ab+b^2`

We can deduce `4x^2 - 32x + 64` to be a perfect square, precisely `color(red)((2x-8)^2)`.

Substitute it into the original expression :

`4x^2 - 32x + 64 - 4 = (2x-8)^2 - 2^2`

That is a difference of squares, so we can factor it out as:

`(2x-8color(red)+ 2)(2x-8color(red)-2)` = `color(blue)((2x-6)(2x-10)`