Bayesian Network - Inference
You're on the right path.  Here's my suggestion.  First, apply the definition of conditional probability:

$$ \Pr[e|a,c,\neg b] = {\Pr[e,a,c,\neg b] \over \Pr[a,c,\neg b]}. $$

So, your job is to compute both $\Pr[e,a,c,\neg b]$ and $\Pr[a,c,\neg b]$.  I suggest that you do each of them separately.

To compute $\Pr[a,\neg b,c,e]$, it is helpful to notice that

$$ \Pr[a,\neg b,c,e] = \Pr[a,\neg b,c,d,e] + \Pr[a,\neg b,c,\neg d,e]. $$

So, if you can compute terms on the right-hand side, then just add them up and you've got  $\Pr[a,\neg b,c,e]$.  You've already computed $\Pr[a,\neg b,c,\neg d,e]$ in part (b).  So, just use the same method to compute $\Pr[a,\neg b,c,d,e]$, and you're golden.

Another way to express the last relation above is to write

$$ \Pr[a,\neg b,c,e] = \sum_d \Pr[a,\neg b,c,D=d,e]. $$

If you think about it, that's exactly the same equation as what I wrote, just using $\sum$ instead of $+$.  You can think about whichever one is easier for you to think about.

Anyway, now you've got $\Pr[e,a,c,\neg b]$.  All that remains is to compute $\Pr[a,c,\neg b]$.  You can do that using exactly the same methods.  I'll let you fill in the details: it is a good exercise.  Finally, plug into the first equation at the top of my answer, and you're done.