jmc

Rewrite the expression as$$4(5x+1)(5x+3)(5x+5)$$Since $x$ is odd, let $x=2n-1$. The expression becomes$$4(10n-4)(10n-2)(10n)=32(5n-2)(5n-1)(5n)$$Consider just the product of the last three terms, $5n-2,5n-1,5n$, which are consecutive. At least one term must be divisible by $2$ and one term must be divisible by $3$ then. Also, since there is the $5n$ term, the expression must be divisible by $5$. Therefore, the minimum integer that always divides the expression must be $32\cdot 2\cdot 3\cdot 5=\boxed{960}$. To prove that the number is the largest integer to work, consider when $x=1$ and $x=5$. These respectively evaluate to be $1920,87360$; their greatest common factor is indeed $960$.