To factor the expression $\sqrt{4x^2 - 3x}$, we should first focus on factoring the expression under the square root, which is $4x^2 - 3x$.

Step 1: Factor $4x^2 - 3x$

The common factor in $4x^2$ and $-3x$ is $x$. Thus, we can factor out $x$:

$4x^2 - 3x = x(4x - 3)$

Step 2: Apply the square root

Now we take the square root of both sides:

$\sqrt{4x^2 - 3x} = \sqrt{x(4x - 3)}$

This can be expressed as:

$\sqrt{4x^2 - 3x} = \sqrt{x} \cdot \sqrt{4x - 3}$

So, the factored form of $\sqrt{4x^2 - 3x}$ is:

$\boxed{\sqrt{x} \cdot \sqrt{4x - 3}}$

Let me know if you'd like more details or have any questions!

Here are 5 related questions:

1. How do you factor quadratic expressions generally?
2. What is the difference between factoring and simplifying an expression?
3. How do you handle square roots in algebraic expressions?
4. What is the importance of factoring in solving equations?
5. Can every polynomial be factored?

Tip: Always look for common factors first when factoring an expression.