The way I would divide it would go as follows:
Write down the first term #color(blue)(2p^3)# of the quotient so that when multiplied by the leading term of the divisor #(color(purple)(2p))# gives the leading term of the dividend #4p^4#.
#(4p^4-17p^2+14p-3) -: (color(purple)(2p-3)) = color(blue)(2p^3)...#
Next note that #(color(purple)(-3))*(color(blue)(2p^3)) = -6p^3# whereas what we want is #0p^3#. So the next term of the quotient is #color(blue)(3p^2)#, which will give us #6p^3# when multiplied by #color(purple)(2p)#, cancelling out the #-6p^3#...
#(4p^4-17p^2+14p-3) -: (color(purple)(2p-3)) = color(blue)(2p^3+3p^2)...#
Next note that #(color(purple)(-3))*(color(blue)(3p^2)) = -9p^2# whereas what we want is #-17p^2#. So the next term of the quotient is #color(blue)(-4p)#, which will give us #-8p^2# when multiplied by #color(purple)(2p)#.
#(4p^4-17p^2+14p-3) -: (color(purple)(2p-3)) = color(blue)(2p^3+3p^2-4p)...#
Next note that #(color(purple)(-3))*(color(blue)(-4p)) = 12p# whereas what we want is #14p#. So the next term of the quotient is #color(blue)(1)#, which will give us #2p# when multiplied by #color(purple)(2p)#.
#(4p^4-17p^2+14p-3) -: (color(purple)(2p-3)) = color(blue)(2p^3+3p^2-4p+1)...#
Finally note that #(color(purple)(-3))*(color(blue)(1)) = -3#, which is exactly what we want, so the division is exact, with no remainder.
#(4p^4-17p^2+14p-3) -: (color(purple)(2p-3)) = color(blue)(2p^3+3p^2-4p+1)#
