Bot GPT 4-o mini To find the product of the polynomials \( (-3x^5 - 4x^4)(7x^2 - 2x + 6) \), we will multiply each term in the first polynomial by each term in the second polynomial.
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Distributing \(-3x^5\):
- \(-3x^5 \cdot 7x^2 = -21x^7\)
- \(-3x^5 \cdot (-2x) = 6x^6\)
- \(-3x^5 \cdot 6 = -18x^5\)
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Distributing \(-4x^4\):
- \(-4x^4 \cdot 7x^2 = -28x^6\)
- \(-4x^4 \cdot (-2x) = 8x^5\)
- \(-4x^4 \cdot 6 = -24x^4\)
Now we combine all the results: \[ (-3x^5 - 4x^4)(7x^2 - 2x + 6) = -21x^7 + 6x^6 - 18x^5 - 28x^6 + 8x^5 - 24x^4 \]
Next, we can combine like terms:
- For \(x^7\): \(-21x^7\)
- For \(x^6\): \(6x^6 - 28x^6 = -22x^6\)
- For \(x^5\): \(-18x^5 + 8x^5 = -10x^5\)
- For \(x^4\): \(-24x^4\)
Putting it all together, the product is: \[ -21x^7 - 22x^6 - 10x^5 - 24x^4 \]
Thus, the final answer is: \[ \boxed{-21x^7 - 22x^6 - 10x^5 - 24x^4} \]
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