I suspect the dividend should have had the term #3x^3# instead of #3x^2#;
however here is the answer to the question as it was asked.
#(color(red)2x^4+color(red)3x^2color(red)(-1)) div (x-color(blue)(1/2))#
#{:
(,,(x^4),(x^3),(x^2),(x^1),(x^0)),
(,"|",color(red)2,0,color(red)3,0,color(red)(-1)),
(underline(" +"color(white)(1/2)),"|",underline(color(white)(1/2" ")),underline(color(blue)(1/2)xxcolor(green)2=1),underline(color(blue)(1/2)xxcolor(orange)(1)=1/2),underline(color(blue)(1/2)xxcolor(brown)(7/2)=7/4),underline(color(blue)(1/2)xxcolor(cyan)(7/4)=7/8)),
(color(blue)(1/2),"|",color(white)(1/2)color(green)(2),color(white)(1/2)color(orange)(1),color(brown)(7/2),color(cyan)(7/4),color(magenta)(-1/8)),
(,,(x^3),(x^2),(x^1),(x^0),"Remainder")
:}#
#(color(red)2x^4+color(red)3x^2color(red)(-1)) div (x-color(blue)(1/2))= color(green)2x^3color(orange)(+1)x^2color(brown)(+7/2)xcolor(cyan)(+7/4)# with a remainder of #(color(magenta)(-1/8))#
