To find the product of the polynomials \( (2x^3 + 3x^2)(4x^4 - 5x^3 - 6x^2) \), we can use the distributive property (also known as the FOIL method for binomials).
Let's compute it step by step:
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Multiply \( 2x^3 \) by 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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Multiply \( 3x^2 \) by each term in \( 4x^4 - 5x^3 - 6x^2 \):
- \( 3x^2 \cdot 4x^4 = 12x^{6} \)
- \( 3x^2 \cdot (-5x^3) = -15x^{5} \)
- \( 3x^2 \cdot (-6x^2) = -18x^{4} \)
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Now, combine all the results:
- The \( x^7 \) term: \( 8x^7 \)
- The \( x^6 \) terms: \( -10x^6 + 12x^6 = 2x^6 \)
- The \( x^5 \) terms: \( -12x^5 - 15x^5 = -27x^5 \)
- The \( x^4 \) term: \( -18x^4 \)
Putting it all together, we get:
\[ 8x^7 + 2x^6 - 27x^5 - 18x^4 \]
Thus, the product of the two polynomials is:
\[ 8x^7 + 2x^6 - 27x^5 - 18x^4 \]
The correct response is: 8x7 + 2x6 - 27x5 - 18x4