How do you divide `(3x^4-3x^3+x^2 + x – 15) /(x – 3)^2`?

Given: `(3x^4 - 3x^3 + x^2 + x - 15)/(x-3)^2`

Using long division, the denominator becomes: `x^2 - 6x + 9`

`x^2 - 6x + 9 | bar(3x^4 - 3x^3 + x^2 + x - 15)`

We want to eliminate `3x^4`. What times `x^2 = 3x^4`? `" "3x^2`

Multiply `3x^2` times each of the terms in the divisor and then subtract:

`" "3x^2`
`x^2 - 6x + 9 | bar(3x^4 - 3x^3 + " "x^2 + x - 15)`
`" "ul(3x^4 - 18x^3 + 27x^2)`
`" "15x^3 - 26x^2`

Bring down the next term from the dividend:

`" "3x^2`
`x^2 - 6x + 9 | bar(3x^4 - 3x^3 + " "x^2 + x - 15)`
`" "ul(3x^4 - 18x^3 + 27x^2)`
`" "15x^3 - 26x^2 + x`

We want to eliminate `15x^3`. What times `x^2 = 15x^3`? `" "15x`

Multiply `15x` times each of the terms in the divisor and then subtract and then bring down the next dividend term:

`" "3x^2 + 15x`
`x^2 - 6x + 9 | bar(3x^4 - 3x^3 + " "x^2 + " "x - 15)`
`" "ul(3x^4 - 18x^3 + 27x^2)`
`" "15x^3 - 26x^2 + " "x`
`" "ul(15x^3 -90x^2 + 135x)`
`" "64x^2 - 134x - 15`

We want to eliminate `64x^2`. What times `x^2 = 64x^2`? `" "64`

Multiply `64` times each of the terms in the divisor and then subtract

`" "3x^2 + 15x + 64`
`x^2 - 6x + 9 | bar(3x^4 - 3x^3 + " "x^2 + " "x - 15)`
`" "ul(3x^4 - 18x^3 + 27x^2)`
`" "15x^3 - 26x^2 + " "x`
`" "ul(15x^3 -90x^2 + 135x)`
`" "64x^2 - 134x - 15`
`" "ul(64x^2 - 384x + 576`
`" "250x - 591`

`(3x^4 - 3x^3 + x^2 + x - 15)/(x-3)^2 = 3x^2+15x+64 + (250x-591)/(x-3)^2`