To find the product of the two polynomials \((2x^3 + 3x^2)(4x^4 - 5x^3 - 6x^2)\), we will distribute each term in the first polynomial to each term in the second polynomial.
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Distributing \(2x^3\):
- \(2x^3 \cdot 4x^4 = 8x^7\)
- \(2x^3 \cdot (-5x^3) = -10x^6\)
- \(2x^3 \cdot (-6x^2) = -12x^5\)
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Distributing \(3x^2\):
- \(3x^2 \cdot 4x^4 = 12x^6\)
- \(3x^2 \cdot (-5x^3) = -15x^5\)
- \(3x^2 \cdot (-6x^2) = -18x^4\)
Next, we combine all of these results: \[ 8x^7 + (-10x^6 + 12x^6) + (-12x^5 - 15x^5) - 18x^4 \]
This simplifies to:
- Combine the \(x^6\) terms: \(-10x^6 + 12x^6 = 2x^6\)
- Combine the \(x^5\) terms: \(-12x^5 - 15x^5 = -27x^5\)
Thus, we have: \[ 8x^7 + 2x^6 - 27x^5 - 18x^4 \]
Therefore, the product of the two polynomials is: \[ \boxed{8x^7 + 2x^6 - 27x^5 - 18x^4} \]