There are two values of a which makes the determinant

$\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a\end{array}\right|=86$

$\Rightarrow 1\left(2 a^{2}+4\right)-2(-4 a-20)=86$

$\Rightarrow 2 a^{2}+4+8 a+40=86$

$\Rightarrow 2 a^{2}+8 a-42=0$

$\Rightarrow a^{2}+4 a-21=0$

$\Rightarrow a^{2}+7 a-3 a-21=0$

$\Rightarrow a(a+7)-3(a+7)=0$

$\Rightarrow(a+7)(a-3)=0$

$\Rightarrow a=-7,3$

Sum of the two values of $a=-7+3=-4$.

Hence, the correct option is (c).