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The given function is:

\[
f(x) = 5x(2x + 3)^3(x^2 - 2)^2(x - 6)^7
\]

Would you like to differentiate this function, expand it, or do something else with it? Let me know your preference, and I can guide you through the solution.

Here are some possible steps:

1. Find the derivative \( f'(x) \) using the product and chain rule.
2. Expand \( f(x) \) (if needed).
3. Evaluate \( f(x) \) at a specific value of \( x \).
4. Analyze the critical points by solving \( f'(x) = 0 \).
5. Simplify or rewrite the expression in a different form.

**Tip:** When differentiating a product of multiple terms, use the product rule carefully, and apply the chain rule to each individual term.

Learn how to differentiate and analyze the polynomial function f(x) = 5x(2x + 3)^3(x^2 - 2)^2(x - 6)^7 using product and chain rules. Explore key concepts in calculus and algebra, including higher-order polynomials and their derivatives. Suitable for advanced high school and college-level students.

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