smc

The mean is $\frac{4+6+8+17+x}{5}=\frac{35+x}{5}$. There are three possibilities for the median: it is either $6$, $8$, or $x$. Let's start with $6$. $\frac{35+x}{5}=6$ has solution $x=-5$, and the sequence is $-5,4,6,8,17$, which does have median $6$, so this is a valid solution. Now let the median be $8$. $\frac{35+x}{5}=8$ gives $x=5$, so the sequence is $4,5,6,8,17$, which has median $6$, so this is not valid. Finally we let the median be $x$. $\frac{35+x}{5}=x⟹35+x=5x⟹x=\frac{35}{4}=8.75$, and the sequence is $4,6,8,8.75,17$, which has median $8$. This case is therefore again not valid. Hence the only possible value of $x$ is $\boxed{-5}.$