To simplify the expression \((4x^3 + 2x^2)(6x - 9)\), we will use the distributive property (also known as the FOIL method for binomials).
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Distribute \(4x^3\) to both terms in the second polynomial \((6x - 9)\): \[ 4x^3 \cdot 6x = 24x^4 \] \[ 4x^3 \cdot (-9) = -36x^3 \]
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Now distribute \(2x^2\) to both terms in the second polynomial: \[ 2x^2 \cdot 6x = 12x^3 \] \[ 2x^2 \cdot (-9) = -18x^2 \]
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Now we combine all the terms: \[ 24x^4 + (-36x^3 + 12x^3) - 18x^2 \]
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Combine like terms: \[ 24x^4 + (-36x^3 + 12x^3) = 24x^4 - 24x^3 \] So, we have: \[ 24x^4 - 24x^3 - 18x^2 \]
Thus, the simplified expression is: \[ \boxed{24x^4 - 24x^3 - 18x^2} \]