To find the product of \((4x^3 + 2x^2)(6x - 9)\), we can use the distributive property (also known as the FOIL method for binomials).
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Distribute each term in the first polynomial by each term in the second polynomial: \[ (4x^3)(6x) + (4x^3)(-9) + (2x^2)(6x) + (2x^2)(-9) \]
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Calculate each of these products:
- \(4x^3 \cdot 6x = 24x^4\)
- \(4x^3 \cdot -9 = -36x^3\)
- \(2x^2 \cdot 6x = 12x^3\)
- \(2x^2 \cdot -9 = -18x^2\)
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Now combine these results: \[ 24x^4 + (-36x^3 + 12x^3) + (-18x^2) \]
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Combine like terms:
- For \(x^3\) terms: \(-36x^3 + 12x^3 = -24x^3\)
Thus, the combined result is: \[ 24x^4 - 24x^3 - 18x^2 \]
Putting this all together, we write the final answer in descending order: \[ \boxed{24x^4 - 24x^3 - 18x^2} \]