Use only the coefficients and fill in #0 " for " 0x^2,# the missing power.
The expression is #4x^3 +0x^2 -5x+15#
If #(x+3)=0" "rArr x =-3", "# write #-3# outside on the left.
STEP 1
#color(white)(www.w.w)|" "4" "0" "-5" "15#
#color(white)(w..wwww)|" "darr#
#color(white)(ww.w)-3|ul(color(white)(wwwwwwwwwwwwwwww)#
#color(white)(www.w.www.w)4color(white)(www.wwwwww.w) larr# bring down the 4
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STEP 2
#color(white)(w.w.w.w)|" "4" "color(red)(0)" "-5" "15#
#color(white)(w..wwww)|#
#color(white)(ww.w)color(blue)(-3)|ul(color(white)
(w..xww)color(blue)(-12))ul(color(white)(wwxxxxwww)larr# multiply #-3xx4#
#color(white)(www.w.wxw.w)color(blue)(4)" "color(red)(-12)color(white)(www.w.w) larrcolor(red)(0)+color(blue)((-12)) = color(red)(-12)#
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STEP 3
#color(white)(www.w.w)|" "4" "0" "color(red)(-5)" "15#
#color(white)(w....wwww)|#
#color(white)(ww.w)color(blue)(-3)|ul(color(white)
(w.ww)-12" "color(blue)(+36))ul(color(white)(ww)larr" "color(blue)(-3xx-12 =36)#
#color(white)(www.w.ww.w)4" "color(blue)(-12)" "color(red)(+31)color(white)(wxxxxw) larrcolor(red)(-5)color(blue)(+36) = color(red)(31)#
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STEP 4
#color(white)(www.w.w)|" "4" "0" "-5" "color(red)(15)#
#color(white)(w....wwww)|#
#color(white)(ww.xw)color(blue)(-3)|ul(color(white)
(w.ww)-12" "+36" "color(blue)(-93))ul(color(white)(ww)larrcolor(blue)(-3xx31)#
#color(white)(www.w.ww.w)4" "-12" "color(blue)(+31)" " color(magenta)(-78)color(white)(wxxw) larrcolor(red)(15)color(blue)(-93)#
These are the coefficients of the answer, starting with #x^2# and the last term #color(magenta)(-78)# is the remainder.
The quotient is #4x^2 -12x+31 " rem" -78#
