Question

How do you factor `4x ^ { 2} + 61x + 180`?

Answer 1

`(4x+45)(x+4)`

Explanation:

`4x^2 +61x +180`

Let's check first if there are exact factors before we waste time looking for them..

`Delta = b^2 -4ac`

(If the answer is a perfect square, then it means there are factors.)

`Delta = 61^2-4(4)(180)= 841 = 29^2`

Now let's look for any clues...

We are looking for factors of `4 and 180` whose products add to `61`

`61` is an odd number, which can only come from `"even +odd"`

So, `2xx2` can be eliminated as possible factors of `4`

Find factor pairs of `180` which have and odd x even.

A bit of trial and error gives us `45 and 4`

`" "4" "45`
`" "darr" "darr`
`" "4" "45" "rarr1xx45" " rarr45`
`" "1" "4" "rarr4xx4 " "rarrul16`
`color(white)(xxxxxxxxxxxxxxx.xx.xx)61`

These are the factors we need,

`4x^2 +61x +180`

`=(4x+45)(x+4)`