Evaluation of #f(x)=color(blue)(1)x^4color(blue)(-4)x^3color(blue)(+0)x^2color(blue)(-4)xcolor(blue)(+6)#

for #f(color(red)(-2))#

Notice the expansion of the expression to include the implied coefficients of #color(blue)(1)# for #x^4# and of #color(blue)(0)# for #x^2#

Initial set-up:

#{:
(,"|",color(blue)(1),color(blue)(-4),color(blue)(+0),color(blue)(-4),color(blue)(+6),color(white)("XXXXXXXX")"line [1]"),
(,"|",,,,,,color(white)("XXXXXXXX")"line[2]"),
("----",,"----","----","----","----","----",),
(xxcolor(red)((-2)),"|",color(green)(1),,,,,color(white)("XXXXXXXX")"line [3]")
:}#

For each column

- Multiply the last number written on line [3] by #color(red)((-2))# and write the product on line [2] of the next column
- Add the numbers in lines [1] and [2] of the next column and write the sum in line [3] of that column.

The number written in the last column of line [3] will be the value of #f(color(red)(-2))#

#{:
(,"|",color(blue)(1),color(blue)(-4),color(blue)(+0),color(blue)(-4),color(blue)(+6),color(white)("XXXXXXXX")"line [1]"),
(,"|",,-2,12,-24,56,color(white)("XXXXXXXX")"line[2]"),
("-----------",,"----","----","----","----","----",),
(xxcolor(red)((-2)),"|",color(green)(1),-6,12,-28,color(cyan)(62),color(white)("XXXXXXXX")"line [3]")
:}#