How do you divide `(-x^4+6x^3+2x^2-7x-7)/(x-2)`?

This requires long division or the shorter method of synthetic division.

set `x-2` equal to `0" "rarr x =2`

Write down the coefficients of the `x` terms in descending order.

`" "2|-1" "6" "2" "-7" "-7`
`" "|ul(" "darrcolor(white)(xxxxxxxxxxxxxxxxxxxxx)`
`color(white)(xxxx)-1" "larr` bring down the `-1`

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`" "color(blue)(2)|-1" "6" "2" "-7" "-7`
`" "|ul(" "darr" "color(blue)(-2)" "larr (color(blue)(2xx-1))" "color(white)(xxxxx)`
`color(white)(xxxxx)color(blue)(-1)" "color(red)(4)" "larr` add `6+(-2)`

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Repeat the process:

Multiply `2` outside by the new value at the bottom:

Multiply `color(red)(2xx4=8)` and then add `2+8=color(lime)(10)`

`" "color(red)(2)|-1" "6" "2" "-7" "-7`
`" "|ul(color(white)(xxxxx)-2" "color(red)(8) color(white)(xxxxxxxxxxxx)`
`color(white)(xxxxx)-1" "color(red)(4)" "color(lime)(10)`

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Multiply `color(lime)(2xx10=20)` and then add `-7+20 = color(magenta)(13)`

`" "color(lime)(2)|-1" "6" "2" "-7" "-7`
`" "|ul(color(white)(xxxxx)-2" "8 " "color(lime)(20) color(white)(xxxxxxxxxxxx)`
`color(white)(xxxxx)-1" "4" "color(lime)(10)" "color(magenta)(13)`

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Multiply `color(magenta)(2xx13=26)` and then add `-7+26 = color(forestgreen)(19)`

`" "color(magenta)(2)|-1" "6" "2" "-7" "-7`
`" "|ul(color(white)(xxxxx)-2" "8 " "20" "color(magenta)(26) color(white)(xxxxxxxxxxxx)`
`color(white)(xxxx)-1" "4" "10" "color(magenta)(13)" "color(forestgreen)(19)" "larr` the remainder

These are the coefficients of the quotient. `(x^4 div x = x^3)`

`=-x^3+4x^2+10x+13" rem " 19`