Symmetric function roots sum

The equation $f(5 + x) = f(5 - x)$ implies that the graph of $f(x)$ is symmetric about the vertical line $x = 5$.

If $r$ is a root of $f(x) = 0$, then $5 + k = r$ for some $k$. Due to symmetry, $5 - k$ must also be a root.

The sum of any pair of roots symmetric about $x = 5$ is $(5 + k) + (5 - k) = 10$.

Since $f(x) = 0$ has 4 distinct real roots, they form 2 such symmetric pairs. Therefore, the sum of all four roots is $2 \times 10 = 20$.