Question

How do you divide `(12x^4+12x^3-5x^2+9x+3)/(x^2-4)`?

Answer 1

The answer is `=12x^2+12x+43+(57x+175)/(x^2-4)`

Explanation:

Let's do a long division

`color(white)(aaaa)` `12x^4` `color(white)(aaaa)` `12x^3` `color(white)(aaaa)` `-5x^2` `color(white)(aaaa)` `9x` `color(white)(aaaa)` `3` `∣` `x^2-4`

`color(white)(aaaa)` `12x^4` `color(white)(aaaaaaaa)` ###color(white)(aaaa)#`-48x^2` `color(white)(aaaa)` ###color(white)(aaaaaa)### `∣` `12x^2+12x+43`

`color(white)(aaaaaa)` `0` `color(white)(aaaaa)` `12x^3` `color(white)(aaa)` ###color(white)(aaaaaaaaaa)#`9x` `color(white)(aaaaaa)`##

`color(white)(aaaaaa)` ###color(white)(aaaaaa)#`12x^3` `color(white)(aaaa)` `+43x^2` `color(white)(aa)` `-48x` `color(white)(aa)` `+3`

`color(white)(aaaaaa)` ###color(white)(aaaaaa)#`0` `color(white)(aaaaaaa)` `+43x^2` `color(white)(aa)` `+57x-172` `color(white)(aaaaaa)`##

`color(white)(aaaaaa)` ###color(white)(aaaaaa)### `color(white)(aaaaaaaaaaaa)` `0` `color(white)(aa)` `+57x+175` `color(white)(aaaaaa)`##

Therefore,

`(12x^4+12x^3-5x^2+9x+3)/(x^2-4)`

`=12x^2+12x+43+(57x+175)/(x^2-4)`