I find this question very tough to crack.
Thanks for your try

Bring the 1680 over the other side, expand, and re-factor. That is one way to do it.

(x-7)(x-3)(x+1)(x+5)-1680=x^{4}-4x^{3}-34x^{2}+76x-1575

Now, factor that.

The Rational Root Theorem may come in handy.

Try dividing it by x-9 to check if 9 is a root. That's because 9 divides into 1575. Upon doing so, if it reduces to a cubic, then that is a factor.

We get:

x^{3}+5x^{2}+11x+175

Now, try factoring that one more time by dividing by x+7. If it reduces to a quadratic, you have it whipped.