How can this bit of boolean logic be simplified? I have greatly simplified down a piece of boolean logic developed from a truth table, but I cannot figure out how to simplify it more. Two of the same variable exist in the different places, which leads me to believe that it can be simplified more.
$ (D \land (A \lor (\lnot C \land B))) \lor (C \land (A \lor \lnot B)) $
How can this be simplified?

What you want to use here is distributivity, 
$$p \wedge (q \vee r) \leftrightarrow (p \wedge q) \vee (p \wedge r)$$
Use this on both the left and right to deduce that your formula is equivalent to
$$(D\wedge A) \vee (D\wedge \neg C \wedge B) \vee (C\wedge A) \vee (C\wedge \neg B)$$
This is called the "disjunctive normal form" of a formula, and it's often one of the two clearest ways to express one-the other is "conjunctive normal form," which you can pass to via a number of applications of de Morgan's laws and distributivity from here, getting
$$(D\vee C) \wedge (A \vee \neg B\vee \neg C)\wedge (B\vee C\vee A)$$
In this case, the conjunctive form has less terms, but which form is simpler eventually comes down to a subjective question.
