The given expression can be simplified step by step by factoring common terms where possible. Let’s analyze and simplify:

Expression:

$-3e^{-\frac{x}{3}}(x^2 + 1) - 18e^{-\frac{x}{3}}x - 9e^{-\frac{x}{3}}$

Step 1: Identify the common factor

The common factor across all terms is: $-3e^{-\frac{x}{3}}$

Step 2: Factor out $-3e^{-\frac{x}{3}}$

Factoring it out from all terms, we get: $-3e^{-\frac{x}{3}} \big[(x^2 + 1) + 6x + 3 \big]$

Step 3: Simplify inside the brackets

Combine like terms: $x^2 + 6x + 1 + 3 = x^2 + 6x + 4$

Thus, the factored expression becomes: $-3e^{-\frac{x}{3}}(x^2 + 6x + 4)$

Final Simplified Expression:

$-3e^{-\frac{x}{3}}(x^2 + 6x + 4)$

This is the simplified version of the given expression.

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Would you like further details, or do you have any questions about this process? 😊

Here are 5 related questions to explore:

1. How can you expand the factored expression back to verify the result?
2. What are the roots of the quadratic equation $x^2 + 6x + 4 = 0$?
3. How does the exponential term $e^{-\frac{x}{3}}$ affect the behavior of the function as $x \to \infty$?
4. Can you determine the critical points of the function?
5. What is the derivative of the simplified expression?

Tip: Always look for common factors when simplifying algebraic expressions—it often makes the work more manageable!