Laws to simplify a boolean formula
You have $(\neg A \wedge \neg C) \vee \text{some_big_expression} $ and you want to show that it's equivalent to $\text{some_big_expression} $.

Simplification rules won't necessarily lead you from the first expression to the second expression. There are systematic ways to test whether two expressions are equivalent, by applying De Morgan's laws in a certain way and other simplification rules (such as $\neg (\neg P) \longrightarrow P$) to transform a formula into equivalent formulas until you reach a disjunctive normal form (or other normal forms, if you apply the laws in a different way). But this doesn't directly get you from one expression to the other.

Another way to prove that two formulas are equivalent is to show that the first implies the second and the second implies the first. Here, it's obvious that 
$\text{some_big_expression}$ implies $(\neg A \wedge \neg C) \vee \text{some_big_expression}$, so the difficulty is to prove the opposite implication: $(\neg A \wedge \neg C) \vee \text{some_big_expression}$ implies $\text{some_big_expression}$. Think about what this means: you want to prove that if $P$ or $Q$ holds then $Q$ holds — in other words, if you already know $Q$, then you don't need to prove $P$, it's already implied by $Q$. This is a consequence of two properties of disjunction, idempotency and monotonicity: if $Q \Rightarrow P$ then $(P \vee Q) \Rightarrow (P \vee P) \Rightarrow P$.

So what's left is to prove that $\neg A \wedge \neg C$ implies $(\neg A \wedge D) \vee (\neg A \wedge B) \vee (\neg B \wedge \neg C)$. I'll let you finish. Hint: the right-hand side has both $B$ and $\neg B$, what can you say about these?

Second exercise: use De Morgan's laws to grind the first formula into the second one.