Dividing one polynomial by another How is this done? For example, how would one simplify the following? 

$$\frac{x^3-12x^2+0x-42}{x^2-2x+1}$$ 

I can do it with long division, but it never makes intuitive sense to me. Either an explanation of the long division algorithm or a new way of solving this would be much appreciated.
First step: By which monomial should you multiply $x^2-2x+1$ in order to get some polynomial with the same leading coefficient as $x^3-12x^2-42$? That's easy: $x$. So, now you do$$x^3-12x^2-42-x\times(x^2-2x+1)=-10x^2-x-42.$$Now, the same question: By which monomial should you multiply $x^2-2x+1$ in order to get some polynomial with the same leading coefficient as $-10x^2-x-42$? Now, the answer is $-10$. So, now you do$$-10x^2-x-42-(-10)\times(x^2-2x+1)=-21x-32.$$Now the degree is less than the degree of $x^2-2x+1$ and therefore there's nothing else to do: the quotient is $x-10$ and the remainder is $-21x-32$.
$$x^3-12x^2-42=x^3-2x^2+x-10x^2+20x-10-21x-32=$$
$$=(x-10)(x-1)^2-21x-32,$$ which says 

$$\frac{x^3-12x^2-42}{(x-1)^2}=x-10+\frac{-21x-32}{(x-1)^2}.$$