Question

How do you factor `x^2+25x+150`?

Answer 1

`x^2+25x+150=(x+10)(x+15)`

Explanation:

We want to look for two numbers. These numbers should have a product of `color(blue)150` and a sum of `color(red)25`.

Lets examine all the possible factors of `150`:

`{:(ul"Product",,,," "ul"Sum"),(color(blue)150=,1xx150,=>,1+150,=151),(color(blue)150=,2xx75,=>,2+75,=77),(color(blue)150=,3xx50,=>,3+50,=53),(color(blue)150=,5xx30,=>,5+30,=35),(color(blue)150=,10xx15,=>,10+15,=color(red)25):}`

So, the pair of numbers we're looking for are `10` and `15`.

So, we see that

`x^2+25x+150=(x+10)(x+15)`

Check this by expanding it again:

`(x+10)(x+15)=x(x+15)+10(x+15)`

`color(white)((x+10)(x+15))=(x^2+15x)+(10x+150)`

`color(white)((x+10)(x+15))=x^2+25x+150`