To find the product of the two polynomials \( (2x^3 + 3x^2)(4x^4 - 5x^3 - 6x^2) \), we will use the distributive property (also known as the FOIL method for binomials, but here we will distribute each term of the first polynomial across each term of the second polynomial):
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Distribute \( 2x^3 \) to each term in \( 4x^4 - 5x^3 - 6x^2 \): \[ 2x^3 \cdot 4x^4 = 8x^{7} \] \[ 2x^3 \cdot (-5x^3) = -10x^{6} \] \[ 2x^3 \cdot (-6x^2) = -12x^{5} \]
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Distribute \( 3x^2 \) to each term in \( 4x^4 - 5x^3 - 6x^2 \): \[ 3x^2 \cdot 4x^4 = 12x^{6} \] \[ 3x^2 \cdot (-5x^3) = -15x^{5} \] \[ 3x^2 \cdot (-6x^2) = -18x^{4} \]
Now, we can combine all these results: \[ 8x^{7} + (-10x^{6} + 12x^{6}) + (-12x^{5} - 15x^{5}) - 18x^{4} \]
- Combine like terms:
- For \( x^7 \): \( 8x^7 \)
- For \( x^6 \): \( -10x^6 + 12x^6 = 2x^6 \)
- For \( x^5 \): \( -12x^5 - 15x^5 = -27x^5 \)
- For \( x^4 \): \( -18x^4 \)
Putting it all together, we have: \[ 8x^7 + 2x^6 - 27x^5 - 18x^4 \]
Thus, the product of the two polynomials \( (2x^3 + 3x^2)(4x^4 - 5x^3 - 6x^2) \) is: \[ \boxed{8x^7 + 2x^6 - 27x^5 - 18x^4} \]
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