The problem is to simplify the given rational expression: $\frac{2u^3 - 12u^2 + 18u}{u^3 - 3u^2}$

1. Problem Description

The problem is to simplify the given rational expression:

$\frac{2u^3 - 12u^2 + 18u}{u^3 - 3u^2}$

2. Solution Steps

First, we factor out the greatest common factor (GCF) from the numerator and the denominator.

The GCF of the numerator $2u^3 - 12u^2 + 18u$ is $2u$.

$2u^3 - 12u^2 + 18u = 2u(u^2 - 6u + 9)$

The GCF of the denominator $u^3 - 3u^2$ is $u^2$.

$u^3 - 3u^2 = u^2(u - 3)$

Now, we rewrite the expression:

$\frac{2u(u^2 - 6u + 9)}{u^2(u - 3)}$

We can factor the quadratic expression in the numerator:

$u^2 - 6u + 9 = (u - 3)(u - 3) = (u - 3)^2$

So, the expression becomes:

$\frac{2u(u - 3)^2}{u^2(u - 3)}$

We can simplify the expression by cancelling common factors. We can cancel a factor of $u$ and a factor of $(u - 3)$.

$\frac{2u(u - 3)(u - 3)}{u \cdot u(u - 3)} = \frac{2(u - 3)}{u}$

Thus, the simplified expression is $\frac{2(u - 3)}{u}$.

3. Final Answer

$\frac{2(u - 3)}{u}$