How do you divide `(-x^4+5x^3-5x^2-6x-2)/(x^2-4)`?

Try as we might, this rational expression just doesn't factor easily. The next step is to try long division because the degree of the numerator is greater than the degree of the denominator.

Express `x^2-4` as the numbers `1color(white)(aa)0color(white)(aa)4`

Express `-x^4+5x^3-5x^2-6x-2` as `-1color(white)(aa)5color(white)(aa)-5color(white)(aa)-6color(white)(aa)2`

This gives us the long division problem

`color(white)(aaaaaaaaa)-1color(white)(aa)5color(white)(aa)-9`
`1color(white)(aa)0color(white)(a)-4|bar(-1color(white)(aa)5color(white)(aa)-5color(white)(a)-6color(white)(aa)2)`
`color(white)(aaa1aaaa)-ul(1color(white)(aa)0color(white)(aaaa)4)color(white)(aaaa)darr color(white)(a.)darr`
`color(white)(aaaaaaaaaaaaa)5color(white)(aa)-9color(white)(aa)-6`
`color(white)(aaaaaaaaaaaaa)ul(5color(white)(aaaa)0color(white)(aa)-20)color(white)(a1)darr`
`color(white)(aaaaaaaaaaaaaaaa)-9color(white)(aaaa)14color(white)(a)-2`
`color(white)(aaaaaaaaaaaaaaaa)ul(-9color(white)(aaaaa)0color(white)(a.a)36)`
`color(white)(aaaaaaaaaaaaaaaaaaaaaaa)14color(white)(a)-38`

So the answer is

`-x^2+5x-9+(14x-38)/(x^2-4)`