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))#