To find the product of the polynomials \((-4x^4 + 10x)(5x^3 - 1)\), we will apply the distributive property (also known as the FOIL method for binomials) by multiplying each term in the first polynomial by each term in the second polynomial.
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Multiply \(-4x^4\) by \(5x^3\): \[ -4x^4 \cdot 5x^3 = -20x^{4+3} = -20x^7 \]
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Multiply \(-4x^4\) by \(-1\): \[ -4x^4 \cdot -1 = 4x^4 \]
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Multiply \(10x\) by \(5x^3\): \[ 10x \cdot 5x^3 = 50x^{1+3} = 50x^4 \]
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Multiply \(10x\) by \(-1\): \[ 10x \cdot -1 = -10x \]
Now, we can combine all of these results: \[ -20x^7 + 4x^4 + 50x^4 - 10x \]
Next, we combine like terms \(4x^4\) and \(50x^4\): \[ 4x^4 + 50x^4 = 54x^4 \]
Putting it all together, we have: \[ -20x^7 + 54x^4 - 10x \]
Thus, the product of the polynomials \((-4x^4 + 10x)(5x^3 - 1)\) is: \[ \boxed{-20x^7 + 54x^4 - 10x} \]