Not an answer, just a long comment.

By making small Manipulate, it shows that no real solution exist when $a$ gets over value of around $-1/10$. Hard to determine the exact value of $a$ but using the slider you can get very close to finding it. i.e when $a$ is larger than that limit, it never crosses the x-axis, so no real solution exist.

```
ClearAll(a,x);
eq = (1 - Exp(x) + (3 x^2)/2)/(2 x) - a;
Manipulate(Plot(eq /. a -> a0, {x, -3, 3}),
{{a0, 0, "a"}, -1, 1, .001, Appearance -> "Labeled"},
TrackedSymbols :> {a0}
)
```

I think this might explain why Solve could not do it in addition to the other comments above.