To find the product of the polynomials \((-4x^4 + 10x)(5x^3 - 1)\), we can use the distributive property (also known as the FOIL method for binomials).
We will distribute each term in the first polynomial \((-4x^4 + 10x)\) with each term in the second polynomial \((5x^3 - 1)\):
- \( -4x^4 \times 5x^3 = -20x^7 \)
- \( -4x^4 \times (-1) = 4x^4 \)
- \( 10x \times 5x^3 = 50x^4 \)
- \( 10x \times (-1) = -10x \)
Now, we combine all these results:
\[ -20x^7 + 4x^4 + 50x^4 - 10x \]
Next, we combine like terms:
\[ -20x^7 + (4x^4 + 50x^4) - 10x = -20x^7 + 54x^4 - 10x \]
So, the product of the polynomials \((-4x^4 + 10x)(5x^3 - 1)\) is:
\[ \boxed{-20x^7 + 54x^4 - 10x} \]