Step 1. Write only the coefficients of #x# in the dividend inside an upside-down division symbol.
#|4" "color(white)(1)7color(white)(1)-13" " "-3#
#|color(white)(1)#
#stackrel("——————————————)#
Step 2. Put the divisor at the left.
#color(red)(-3)|4" "color(white)(1)7color(white)(1)-13" " "-3#
#" "color(white)(1)|#
#" "color(white)(1)stackrel("——————————————)#
Step 3. Drop the first coefficient of the dividend below the division symbol.
#-3|4" "color(white)(1)7color(white)(1)-13" " "-3#
#" "color(white)(1)|#
#" "color(white)(1)stackrel("——————————————)#
#" "color(white)(1)|color(red)(4)#
Step 4. Multiply the result by the constant, and put the product in the next column.
#-3|4" "color(white)(1)7color(white)(1)-13" " "-3#
#" "color(white)(1)|color(white)(1)color(red)(-12)#
#" "color(white)(1)stackrel("——————————————)#
#" "color(white)(1)|4#
Step 5. Add down the column.
#-3|4" "color(white)(1)7color(white)(1)-13" " "-3#
#" "color(white)(1)|-12#
#" "color(white)(1)stackrel("——————————————)#
#" "color(white)(1)|4color(white)(1)color(red)(-5)#
Step 6. Repeat Steps 4 and 5 until you can go no farther.
#-3|4" "color(white)(1)7color(white)(1)-13" " "-3#
#" "color(white)(1)|-12" "color(white)(1)15" "color(white)(1)-6#
#" "stackrel("——————————————)#
#color(blue)(" "color(white)(1)|4-5" "" "2)" "color(white)(1)color(red)(-9)#
#(4x^3+7x^2-13x-3)/(x+3) = 4x^2-5x+2-9/(x+3)#
Check:
#(x+3)(4x^2-5x+2-9/(x+3)) = (x+3)(4x^2-5x+2)-9#
#= 4x^3-5x^2+2x+12x^2-15x+6-9 = 4x^3+7x^2-13x-3#
