Let's do the long division
#color(white)(aaaa)##5x^4-3x^3+2x^2-1##color(white)(aaaaaaaa)##|##x^2+4#
#color(white)(aaaa)##5x^4##color(white)(aa aaaa)##+20x^2##color(white)(aaaaaaaaaaa)##|##5x^2-3x-18#
#color(white)(aaaa)##0-3x^3##color(white)(aa a)##-18x^2##color(white)(aaaaaaa)#
#color(white)(aaaaaa)##-3x^3##color(white)(aaaa)##-18x^2-12x-1##color(white)(aaaaaaa)#
#color(white)(aaaaaaaa)##0##color(white)(aaaaa)##-18x^2-12x##color(white)(a)##-72#
#color(white)(aaaaaaaaaaaaaaaaa)##0##color(white)(aaa)##+12x##color(white)(aa)##+71#
Therefore,
#(5x^4-3x^3+2x^2-1)/(x^2+4)=5x^2-3x-18+(12x+71)/(x^2+4)#
The quotient is #=5x^2-3x-18#
The remainder is #=12x+71#