To find the product of \((4x^3 + 2x^2)(6x - 9)\), we will use the distributive property (also known as the FOIL method for binomials).
Let's distribute each term in the first polynomial \((4x^3 + 2x^2)\) to each term in the second polynomial \((6x - 9)\):
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Multiply \(4x^3\) by \(6x\): \[ 4x^3 \cdot 6x = 24x^4 \]
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Multiply \(4x^3\) by \(-9\): \[ 4x^3 \cdot (-9) = -36x^3 \]
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Multiply \(2x^2\) by \(6x\): \[ 2x^2 \cdot 6x = 12x^3 \]
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Multiply \(2x^2\) by \(-9\): \[ 2x^2 \cdot (-9) = -18x^2 \]
Now, we can combine all the terms we obtained: \[ 24x^4 - 36x^3 + 12x^3 - 18x^2 \]
Next, we combine the like terms for \(x^3\): \[ -36x^3 + 12x^3 = -24x^3 \]
So, we have: \[ 24x^4 - 24x^3 - 18x^2 \]
Putting it all together, the final expression in descending order of exponents is: \[ \boxed{24x^4 - 24x^3 - 18x^2} \]