The given expression is:

$\frac{4x^2 - 4x}{x - 1}$

We can simplify this expression step by step:

Step 1: Factor the numerator

The numerator $4x^2 - 4x$ has a common factor of $4x$, so we can factor it out:

$4x^2 - 4x = 4x(x - 1)$

Step 2: Substitute into the expression

Now, the expression becomes:

$\frac{4x(x - 1)}{x - 1}$

Step 3: Simplify the expression

We can cancel out the common factor $(x - 1)$ in both the numerator and denominator, provided $x \neq 1$ (to avoid division by zero):

$\frac{4x(x - 1)}{x - 1} = 4x \quad \text{(for } x \neq 1\text{)}$

Thus, the simplified expression is:

$4x$

Conclusion:

The simplified form of the expression is $4x$, as long as $x \neq 1$.

Would you like more details or have any questions?

Here are some related questions for practice:

1. Simplify $\frac{x^2 - 3x}{x - 3}$.
2. Factor and simplify $\frac{5x^2 - 10x}{x^2 - 2x}$.
3. Simplify $\frac{x^3 - x}{x - 1}$.
4. Simplify $\frac{6x^2 - 18x}{3x - 9}$.
5. Factor and simplify $\frac{2x^2 + 4x}{x + 2}$.

Tip: Always check for factors common to both the numerator and denominator before simplifying an algebraic expression.