$ (\lnot q \lor r) \land ( q \lor \lnot r)\tag{1}$ $[(\lnot q \lor r) \land q] \lor [(\lnot q \lor r) \land \lnot r]\tag{2}$ $[(\lnot q \land q) \lor (r \land q)] \lor [(\lnot q \land \lnot r) \lor (r \land \lnot r)]\tag{3}$ $ \text{False} \lor (r \land q) \lor (\lnot q \land \lnot r) \lor \text{False}\tag{4}$
$(q \land r) \lor (\lnot q \land \lnot r)\tag{5}$
Note that $(5)$ is is in disjunctive normal form (DNF), and is equivalent to $(1)$.
$(1) \to (2)$: Distribution;
$(2) \to (3)$: Distribution, twice;
$(3) \to (4)$: Contradictions $\lnot q \land q \rightarrow \text{False}$ and $\lnot r \land r\rightarrow \text{False}$;
$(4) \to (5)$: Simplification (removal of contradictory disjuncts).
Observation: both $(1)$ and $(5)$ are equivalent to $q \leftrightarrow r$:
To see this, you can rewrite $(1)$ as: $(\lnot q \lor r) \land (\lnot r \lor q) \iff (q \rightarrow r) \land (r\rightarrow q) \iff (q \leftrightarrow r).$
$q \leftrightarrow r\; \text{ is true if and only if }\;(q \land r) \text{ is true or}\; (\lnot q \land \lnot r)\text{ is true.}$ $\text{That is,}\;q \leftrightarrow r\; \text{ is true if and only if }\;\;(q \land r) \lor (\lnot q \land \lnot r).\tag{5} $
p | q |$\;\lnot q \lor r$ | $q \lor \lnot r\;$|$\;(\lnot q \lor r) \land (q\lor \lnot r)$
T | T | $\quad$T $\quad$|$\quad$ T $\quad$| $\quad\quad\quad\quad$T$\quad\quad\leftarrow\quad\;\; (q \land r)$
T | F | $\quad$F $\quad$|$\quad$ T $\quad$| $\quad\quad\quad\quad$F
F | T | $\quad$T $\quad$|$\quad$ F $\quad$| $\quad\quad\quad\quad$F
F | F | $\quad$T $\quad$|$\quad$ T $\quad$| $\quad\quad\quad\quad$T$\quad\quad\leftarrow\quad (\lnot q \land \lnot r)$