First, [tex]x^3=x^2\cdot x[/tex], and if we multiply [tex]x+3[/tex] by [tex]x^2[/tex] we get
[tex]x^3+3x^2[/tex]
Subtracting this from the numerator gives a remainder of
[tex]-2x^2-2x+14[/tex]
Next, [tex]-2x^2=-2x\cdot x[/tex], and if we multiply [tex]x+3[/tex] by [tex]-2x[/tex] we have
[tex]-2x^2-6x[/tex]
and subtracting this from the previous remainder, we end up with a new remainder of
[tex]4x+14[/tex]
Next, [tex]4x=4\cdot x[/tex], and if we multiply [tex]x+3[/tex] by [tex]4[/tex] we get
[tex]4x+12[/tex]
Subtracting from the previous remainder, we get a new remainder of
[tex]2[/tex]
which contains no more factors of [tex]x[/tex], so we're done.
So,
[tex]\dfrac{x^3+x^2-2x+14}{x+3}=x^2-2x+4+\dfrac2{x+3}[/tex]
