The expression you've provided is:

$(3 - x) \cdot \sqrt{\frac{x^2 + 3}{(x - 3)^3}}$

Let's simplify and analyze this step-by-step.

1. Factor the term $3 - x$:

   Notice that $3 - x = -(x - 3)$. So we can rewrite the expression as:

   $-(x - 3) \cdot \sqrt{\frac{x^2 + 3}{(x - 3)^3}}$

2. Combine the terms involving $(x - 3)$:

   The expression now becomes:

   $-\frac{(x - 3)}{\sqrt{(x - 3)^3}} \cdot \sqrt{x^2 + 3}$

   Since $\sqrt{(x - 3)^3} = (x - 3)^{3/2}$, we simplify the expression further:

   $-\frac{(x - 3)}{(x - 3)^{3/2}} \cdot \sqrt{x^2 + 3}$

3. Simplify the powers of $(x - 3)$:

   The powers of $(x - 3)$ simplify as:

   $\frac{(x - 3)}{(x - 3)^{3/2}} = \frac{1}{(x - 3)^{1/2}} = \frac{1}{\sqrt{x - 3}}$

   So the expression becomes:

   $-\frac{\sqrt{x^2 + 3}}{\sqrt{x - 3}}$

4. Final simplified expression:

   The final simplified form of the given expression is:

   $-\sqrt{\frac{x^2 + 3}{x - 3}}$

This is the simplified version of the expression.

Would you like further details or have any questions?

Related Questions:

1. How can I rationalize the denominator in the final result?
2. What happens if $x = 3$? Is the expression undefined?
3. Can this expression be integrated or differentiated? How would you approach it?
4. What if $(x - 3)$ were squared instead of cubed? How would the process change?
5. How would this expression change if we swapped $3 - x$ for $x - 3$?

Tip:

Always check for values that make the denominator zero to identify potential points of discontinuity in an expression.