Question

How do you factor `45r^2-120rs+80s^2`?

Answer 1

`45r^2-120rs+80s^2 = 5(3r-4s)^2`

Explanation:

First separate out the common scalar factor `5`:

`45r^2-120rs+80s^2=5(9r^2-24rs+16s^2)`

Next note that both `9r^2 = (3r)^2` and `16s^2 = (4s)^2` are perfect squares. So given the middle minus sign check to see if this is the square of `(3r-4s)`:

`(3r-4s)^2 = (3r)^2-2(3r)(4s)+(4s)^2 = 9r^2-24rs+16s^2`

So `9r^2-24rs+16s^2` is a perfect square trinomial.