Find the sum of the values of x for which this expression is a minimum.

$(x+1)(x+3)(x+5)(x+7)+2020$ is minimum when $(x+1)(x+3)(x+5)(x+7)$ is minimum. Let

$\begin{aligned} P & = (x+1)(x+3)(x+5)(x+7) & \small \blue{\text{Let }u = x+4} \\ & = (u-3)(u-1)(u+1)(u+3) \\ & = (u^2 - 9)(u^2 - 1) \\ & = u^4 - 10u^2 + 9 \\ & = (u^2 - 5)^2 - 16 \end{aligned}$

Since $(u^2 - 5)^2 \ge 0$ , $P$ is minimum, when $(u^2 - 5)^2 = 0$ or

$\begin{aligned} (x+4)^2 - 5 & = 0 \\ x^2 + 8x + 11 & = 0 \end{aligned}$

By Vieta's formula , the sum of $x$ 's when the expression is minimum is $\boxed{-8}$ .