How do you divide #(2x^4 -x^3 - 9x^2 - x + 5)/ (5x^2 - 2) # using polynomial long division?

Given: #(color(brown)(2x^4-x^3-9x^2-x+5))/(color(green)(5x^2-2))#

Using place keepers of no value. Example: #0x^3#

#color(white)("ddddddddddddddd")color(brown)(2x^4-color(white)("d")x^3color(white)("d")-9x^2-x+5)#
#color(magenta)(+2/5x^2)color(green)((5x^2-2)) ->ul(2x^4+0x^3-4/5x^2larr" Subtract")#
#color(white)("ddddddddddddddddd")0-x^3-41/5x^2-x+5 #
#color(magenta)(-1/5)color(green)((5x^2-2)) ->color(white)("d.dd")ul( -x^3+color(white)("d.")0x^2+2/5xlarr" Subtract")#
#color(white)("ddddddddddddddddddddd")0-41/5x^2-7/5x+5 #
#color(magenta)(-41/25)color(green)((5x^2-2)) ->color(white)("dddddddd")ul(-41/5x^2+0x+81/25larr" Sub.")#
#color(white)("d")color(magenta)("Remainder "->color(white)("ddddddddddddd")0-7/5x+44/25)#

#color(magenta)("Set remainder as: "(-35x+44)/25)# giving:

#color(magenta)(2/5x^2-1/5x-41/25+[(-35x+44)/25 color(green)(-:(5x^2-2))])#

#color(magenta)(2/5x^2-1/5x-41/25+[(-35x+44)/(25 color(green)((5x^2-2)))])#

Perform the polynomial long division

#color(white)(aaaa)##2x^4-x^3-9x^2-x+5##color(white)(aaaa)##|##5x^2-2#

#color(white)(aaaaaaaaaaaaaaaaaaaaaaaa)##color(white)(aaaa)##|##2/5x^2-x/5-41/25#

#color(white)(aaaa)##2x^4-0x^3-4/5x^2#

#color(white)(aaaa)##0x^4-1x^3-41/5x^2-x#

#color(white)(aaaaaaaa)##-1x^3-00x^2+2/5x#

#color(white)(aaaaaaaa)##-0x^3-41/5x^2-7/5x+5#

#color(white)(aaaaaaaaaaaaa)##-41/5x^2-7/5x+82/25#

#color(white)(aaaaaaaaaaaaaaa)##-0x^2-7/5x+43/25#

Therefore,

#(2x^4-x^3-9x^2-x+5)/(5x^2-2)=(2/5x^2-x/5-41/25)+(-7/5x+43/25)/(5x^2-2)#

The remainder is #=(-7/5x+43/25)# and the quotient is #=(2/5x^2-x/5-41/25)#