How do you find the quotient of (x^3 + 4x -7) by x-3?

`"one way is to use the divisor as a factor in the numerator"`

`"consider the numerator"`

`color(red)(x^2)(x-3)color(magenta)(+3x^2)+4x-7`

`=color(red)(x^2)(x-3)color(red)(+3x)(x-3)color(magenta)(+9x)+4x-7`

`=color(red)(x^2)(x-3)color(red)(+3x)(x-3)color(red)(+13)(x-3)color(magenta)(+39)-7`

`=color(red)(x^2)(x-3)color(red)(+3x)(x-3)color(red)(+13)(x-3)+32`

`"quotient "=color(red)(x^2+3x+13)," remainder "=32`

Note that I am using a place keepers `0x^2`. It has no value.

`color(white)("dddddddd.ddd.ddd")x^3+0x^2+4x-7`
`color(magenta)(+x^2)color(green)((x-3))->color(white)("ddd") ul(x^3-3x^2 larr" Subtract")`
`color(white)("ddddddddddddddd")0color(white)("d")+3x^2+4x-7`
`color(magenta)(+3x)color(green)((x-3))->color(white)("ddddddd") ul(3x^2-9x larr" Subtract")`
`color(white)("ddddddddddddddddddd") 0color(white)("d")+13x-7`
`color(magenta)(+13)color(green)((x-3))->color(white)("ddddddddddd")ul(13x-39 larr" Subtract")`
`color(white)("dddddddddddddddddddddddd")0color(white)("d")color(magenta)(+32 larr" Remainder")`

`color(magenta)(x^2+3x+13+32/color(green)((x-3))`