To find the product of the polynomials \((-3x^5 - 4x^4)(7x^2 - 2x + 6)\), we will use the distributive property (also known as the FOIL method for binomials), multiplying each term in the first polynomial by each term in the second polynomial.
Let's distribute:
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Multiply \(-3x^5\) by each term in \((7x^2 - 2x + 6)\):
- \(-3x^5 \cdot 7x^2 = -21x^7\)
- \(-3x^5 \cdot (-2x) = 6x^6\)
- \(-3x^5 \cdot 6 = -18x^5\)
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Next, multiply \(-4x^4\) by each term in \((7x^2 - 2x + 6)\):
- \(-4x^4 \cdot 7x^2 = -28x^6\)
- \(-4x^4 \cdot (-2x) = 8x^5\)
- \(-4x^4 \cdot 6 = -24x^4\)
Now, combine all the obtained terms:
\[ -21x^7 + 6x^6 - 18x^5 - 28x^6 + 8x^5 - 24x^4 \]
Next, combine like terms:
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Combine \(6x^6\) and \(-28x^6\): \[ 6x^6 - 28x^6 = -22x^6 \]
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Combine \(-18x^5\) and \(8x^5\): \[ -18x^5 + 8x^5 = -10x^5 \]
So the resulting polynomial is:
\[ -21x^7 - 22x^6 - 10x^5 - 24x^4 \]
Thus, the product of the polynomials \((-3x^5 - 4x^4)(7x^2 - 2x + 6)\) is:
\[ \boxed{-21x^7 - 22x^6 - 10x^5 - 24x^4} \]