#(color(red)4x^2+color(red)1x+color(red)1)div(x-2)# using synthetic division.
Set up using the zero of the dividend #(x-2)#, or #x=color(blue)2#, and the coefficients #color(red)(4 color(white)(a)1 color(white)(a)1)# of the dividend.
#color(blue)2|color(red)4color(white)(aaa)color(red)1color(white)(aaa)color(red)1#
#color(blue)2|color(red)4color(white)(aaa)color(red)1color(white)(aaa)color(red)1#
#color(white)(aa)darr#
#color(white)(a^2a)color(magenta)4color(white)(aaaaaaaaaaa)#Pull down the 4
#color(blue)2|color(red)4color(white)(aaa)color(red)1color(white)(aaa)color(red)1#
#color(white)(aa)darrcolor(white)(a^11)color(limegreen)8color(white)(aaaaaaa)#Multiply #color(blue)2 xx color(magenta)4=color(limegreen)8#
#color(white)(a^2a)color(magenta)4color(white)(aaaaaaaaa)#
#color(blue)2|color(red)4color(white)(aaa)color(red)1color(white)(aaa)color(red)1#
#color(white)(aa)darrcolor(white)(a^11)color(limegreen)8#
#color(white)(a^2a)color(magenta)4color(white)(aaa)color(magenta)9color(white)(aaaaaaaa)#Add #color(red)1 + color(limegreen)8=color(magenta)9#
#color(blue)2|color(red)4color(white)(aaa)color(red)1color(white)(aaa)color(red)1#
#color(white)(aa)darrcolor(white)(a^11)color(limegreen)8color(white)(aa)color(limegreen)(18)color(white)(aaa)#Multiply #color(blue)2 xx color(magenta)9=color(limegreen)(18)#
#color(white)(a^2a)color(magenta)4color(white)(aaa)color(magenta)9color(white)(aaaaaaaa)#
#color(blue)2|color(red)4color(white)(aaa)color(red)1color(white)(aaa)color(red)1#
#color(white)(aa)darrcolor(white)(a^11)color(limegreen)8color(white)(aa)color(limegreen)(18)color(white)(aaa)#
#color(white)(a^2a)color(magenta)4color(white)(aaa)color(magenta)9color(white)(aa)color(magenta)(19)color(white)(aaaa)#Add #color(red)1+color(limegreen)(18)=color(magenta)(19)#
The #color(magenta)4# and the #color(magenta)9# represent the coefficients of the quotient. The #color(magenta)(19)# is the numerator of the remainder.
The quotient is then #color(magenta)4x+color(magenta)(9)+frac{color(magenta)19]{x-2]#