Exponents!

$a^{2x} - (a^3 + a) \cdot a^{x-1} + a^2 = 0$

$\Rightarrow a^{2x} - a(a^2 + 1) a^x \cdot a^{-1} + a^2 = 0$

$\Rightarrow a^{2x} - (a^2 + 1) a^x + a^2 = 0$

$\Rightarrow p^2 - (a^2 + 1) p+a^2 = 0$ $~ ~~~ ~~ ~~ ~~~~~$ $;[a^x = p]$

$\Rightarrow p^2 -a^2p-p + a^2 = 0$

$\Rightarrow (p-1)(p-a^2) =0$

$p = 1$

Solving for $x$

$\Rightarrow a^x = 1 = a^0$

$x = 0$

Or,

$p = a^2$

$a^x = a^2$

$x = 2$

So, $x = 0,2$

Hence the smallest value of $x$ is $\boxed{0}$