Division by synthetic division is the simplest and quickest method in this case. We will only use the numerical coefficients.
#x-3 =0 " "rarr x =3" "# goes outside on the left.
#(1x^3 +1x^2 +3x+1)/(x-3)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#" "3|" "1" "1" "3" "1#
#" "|ul" "darrul" "#
#" "1color(white)(xxxxxxxxxxxxxxx)larr# bring down the #1#
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#" "color(red)(3)|" "1" "1" "3" "1#
#" "|ul" "darrul" "color(blue)(3)ulcolor(white)(xxxxxxxxxxx)larrcolor(red)(3xx1) = color(blue)(3)#
#" "color(red)(1)" "color(blue)(4)color(white)(xxxxxxxxxxx)larr 1+color(blue)(3 =4)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#" "color(red)(3)|" "1" "1" "3" "1#
#" "|ul" "darrul" "3ul" "color(green)(12)ulcolor(white)(xxxxxxxxx)larrcolor(red)(3xxcolor(blue)(4) color(green)(=12)#
#" "1" "color(blue)(4)" "color(green)(15)color(white)(xxxxxxxxx)larr 3+color(green)(12 =15)#
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#" "color(red)(3)|" "1" "1" "3" "1#
#" "|ul" "darrul" "3ul" "12" "color(magenta)(45)ulcolor(white)(xxxxxx)larrcolor(red)(3xxcolor(green)(15) color(magenta)(=45)#
#" "1" "color(blue)(4)" "color(green)(15)" "color(magenta)(46)color(white)(xxxxxx)larr 1+color(magenta)(45 =46)#
#color(white)(xxxxxxxxxxxxxxxx)uarr#
#color(white)(xxxxxxxxxxxxxxx)"remainder"#
The bottom line gives the numerical coefficients of the quotient.
#x^3 divx =x^2#
Quotient:
#1x^2 +color(blue)(4)x +color(green)15" rem " color(magenta)(46)#