To find the product of the polynomials \((-3x^5 - 4x^4)(7x^2 - 2x + 6)\), we'll use the distributive property (also known as the FOIL method for two binomials, though this is a trinomial and a polynomial).
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Distribute \(-3x^5\) across each term of \((7x^2 - 2x + 6)\): \[ -3x^5 \cdot 7x^2 = -21x^7 \] \[ -3x^5 \cdot (-2x) = 6x^6 \] \[ -3x^5 \cdot 6 = -18x^5 \]
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Distribute \(-4x^4\) across each term of \((7x^2 - 2x + 6)\): \[ -4x^4 \cdot 7x^2 = -28x^6 \] \[ -4x^4 \cdot (-2x) = 8x^5 \] \[ -4x^4 \cdot 6 = -24x^4 \]
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Now combine all of the resulting terms: \[ -21x^7 + 6x^6 - 18x^5 - 28x^6 + 8x^5 - 24x^4 \]
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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 \(x^4\): \(-24x^4\)
Thus, the final result is: \[ -21x^7 - 22x^6 - 10x^5 - 24x^4 \]
So, the product of the polynomials \((-3x^5 - 4x^4)(7x^2 - 2x + 6)\) is: \[ \boxed{-21x^7 - 22x^6 - 10x^5 - 24x^4} \]
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