Let $P(x) = (x - a)(x - b)(x - c)(x - d)(x - e)$. Then the polynomial in question is just $P'(x)$. Note that since $a, b, c, d, e$ are all distinct, on the graph of $P(x)$ between every pair of two consecutive roots there is a local maximum/minimum (where the derivative is zero). But there are four such pairs, so there are at least four real roots of $P'(x)$. Since $P'(x)$ has degree four, there are four distinct real solutions (they are distinct easily since if $a < m < b$ and $b < n < c$ then $m \not= n$).