How to find a substitution that transforms an equation into a particular equation. (Roots of polynomials). Find the value of $c$ so that the substitution $x=u+c$ transforms the equation $x^3-12x^2+45x-54=0$ into the equation $u^3-3u^2=0$
My first idea is to find the roots of $u^3-3u^2=0$ And getting $u=0$ and $u=3$, however I am not sure how to continue.
If someone could explain how to tackle this sort of problem, I would greatly appreciate it. Thanks.

You want a factorization of the form $u^2(u-3)$. This means, the given polynomial should also factor into $(x-c)^2(x-c-3)$.
So, the first thing to do, whether you can factor the given polynomial.
Based on the rational root test you find quickly that your polynomial factors into 
$$(x-3)^2(x-6) \Rightarrow u = x-3 \Leftrightarrow x= u+3$$
