To simplify the expression $3x^2(4x^3 - 2x^2 + 3x - 5)$, we'll distribute $3x^2$ to each term inside the parentheses.

Step-by-Step Solution:

1. Distribute $3x^2$ to each term inside the parentheses:

   $3x^2 \cdot 4x^3 = 12x^5$ $3x^2 \cdot (-2x^2) = -6x^4$ $3x^2 \cdot 3x = 9x^3$ $3x^2 \cdot (-5) = -15x^2$

2. Combine all terms to write the final expression:

   $12x^5 - 6x^4 + 9x^3 - 15x^2$

Final Answer:

$12x^5 - 6x^4 + 9x^3 - 15x^2$

Would you like more details on this solution, or do you have any questions?

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Here are five related questions to explore further:

1. How would the result change if $x^2$ was replaced with $x^3$ outside the parentheses?
2. What if the polynomial inside the parentheses had an extra term, say $+7$?
3. How would we simplify $3x(4x^3 - 2x^2 + 3x - 5)$ without the $x^2$?
4. Can we apply similar steps if this were a division problem instead?
5. What happens if we factor out common terms from the simplified expression?

Tip: When distributing, be careful with signs; any negative terms in the parentheses will reverse signs in each product.