[tex]\dfrac{x^3+10x^2+13x+39}{x^2+2x+1}[/tex]
[tex]x^3=x\cdot x^2[/tex], and [tex]x(x^2+2x+1)=x^3+2x^2+x[/tex]. Subtracting this from the numerator gives a remainder of
[tex](x^3+10x^2+13x+39)-(x^3+2x^2+x)=8x^2+12x+39[/tex]
[tex]8x^2=8\cdot x^2[/tex], and [tex]8(x^2+2x+1)=8x^2+16x+8[/tex]. Subtracting this from the previous remainder gives a new remainder of
[tex](8x^2+12x+39)-(8x^2+16x+8)=-4x+31[/tex]
[tex]-84x[/tex] is not a multiple of [tex]x^2[/tex], so we're done. Then
[tex]\dfrac{x^3+10x^2+13x+39}{x^2+2x+1}=x+8+\dfrac{-4x+31}{x^2+2x+1}[/tex]
