Odtnhj

Premise:
(Tet(a) ^ Tet(b)) v (Cube(c) ^ Cube(d))

Cube(c) -> Dodec(e)

Goal:
~Tet(a) -> Dodec(e)

Anyone have a clue on where to start with this?

I agree with Graham Kemp's "skeleton for the proof".

Rather than provide a skeleton, I will provide a completed proof but using a different proof checker. To make this work in the proof checker I renamed the statements.

You may not be able to use all of the inference rules as they are used here. I used conjunction elimination (∧E), contradiction introduction (⊥I), explosion (X), conditional introduction (→I), and disjunction elimination (∨E).

Klement's proof checker and information about the rules I used can be found in forall x referenced below.

Reference

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

Clearly you want a Conditional Proof to prove that conditional. Assume ~Tet(a) aiming to derive Dodec(e).

Now look at the to premises and the assumption and ask: how may I derive Dodec(e) from that disjunction, conditional, and negation?

| (Tet(a) ^ Tet(b)) v (Cube(c) ^ Cube(d)) Premise
|_ Cube(c) -> Dodec(e) Premise
| |_ ~Tet(a) Assume
| | :
| | Dodec(e) 
| ~Tet(a) -> Dodec(e) Conditional Introduction