[tex]4x^5=x^2\cdot4x^3[/tex], and [tex]4x^3(x^2-2x)=4x^5-8x^4[/tex]. Subtract this from [tex]4x^5-3x^3+2x-1[/tex] to get a remainder of
[tex](4x^5-3x^3+2x-1)-(4x^5-8x^4)=8x^4-3x^3+2x-1[/tex]
[tex]8x^4=x^2\cdot8x^2[/tex], and [tex]8x^2(x^2-2x)=8x^4-16x^3[/tex]. Subtract this from the previous remainder to get a new remainder of
[tex](8x^4-3x^3+2x-1)-(8x^4-16x^3)=13x^3+2x-1[/tex]
[tex]13x^3=x^2\cdot13x[/tex], and [tex]13x(x^2-2x)=13x^3-26x^2[/tex]. Subtract this from the previous remainder to get a new remainder of
[tex](13x^3+2x-1)-(13x^3-26x^2)=26x^2+2x-1[/tex]
[tex]26x^2=x^2\cdot26[/tex], and [tex]26(x^2-2x)=26x^2-52x[/tex]. Subtract this from the previous remainder to get a new remainder of
[tex](26x^2+2x-1)-(26x^2-52x)=54x-1[/tex]
[tex]x^2[/tex] does not divide [tex]54x[/tex], so we're done, and we've found that
[tex]\dfrac{4x^5-3x^3+2x-1}{x^2-2x}=4x^3+\dfrac{8x^4-3x^3+2x-1}{x^2-2x}[/tex]
[tex]\dfrac{4x^5-3x^3+2x-1}{x^2-2x}=4x^3+8x^2+\dfrac{13x^3+2x-1}{x^2-2x}[/tex]
[tex]\dfrac{4x^5-3x^3+2x-1}{x^2-2x}=4x^3+8x^2+13x+\dfrac{26x^2+2x-1}{x^2-2x}[/tex]
[tex]\dfrac{4x^5-3x^3+2x-1}{x^2-2x}=\boxed{4x^3+8x^2+13x+26+\dfrac{54x-1}{x^2-2x}}[/tex]
