To find the product of the polynomials \( (4x^3 + 2x^2)(6x - 9) \), we will use the distributive property (also known as the FOIL method in the case of binomials).
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Distribute \( 4x^3 \) to both terms in \( (6x - 9) \): \[ 4x^3 \cdot 6x = 24x^4 \] \[ 4x^3 \cdot (-9) = -36x^3 \]
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Distribute \( 2x^2 \) to both terms in \( (6x - 9) \): \[ 2x^2 \cdot 6x = 12x^3 \] \[ 2x^2 \cdot (-9) = -18x^2 \]
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Now, combine all the terms we obtained: \[ 24x^4 + (-36x^3 + 12x^3) - 18x^2 \] \[ = 24x^4 + (-36x^3 + 12x^3) - 18x^2 \] \[ = 24x^4 - 24x^3 - 18x^2 \]
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Finally, write the expression in descending order of exponents: \[ \boxed{24x^4 - 24x^3 - 18x^2} \]