It looks like the problem statement is written in black, and your work is in red. So the problem statement is:
\(\displaystyle \left(P \vee \neg(\neg(\neg Q)) \right) \equiv \left( \neg Q \wedge P \right)\)
Assuming this is the case, your answer is correct, although your justification is wrong. On the left-hand side, two of the negations cancel out and then some parentheses collapse. Everything you did is fine up until this step:
\(\displaystyle \left( P \vee \neg Q \right) \equiv \left(\neg Q \wedge P \right)\)
Your final step is incorrect because:
\(\displaystyle \left( P \vee \neg Q \right) \not\equiv \left( P \wedge Q \right)\)
We can easily see this equivalence does not hold by testing some values. Suppose P is true and Q is false. Then we'd have:
\(\displaystyle \left( \text{TRUE} \vee \neg \text{FALSE} \right) \stackrel{?}{\equiv} \left( \text{TRUE} \wedge \text{FALSE} \right)\)
\(\displaystyle \left( \text{TRUE} \vee \text{TRUE} \right) \stackrel{?}{\equiv} \text{FALSE}\)
\(\displaystyle \text{TRUE} \not\equiv \text{FALSE}\)
Similarly, we can see the equivalence used on the right-hand side doesn't hold either. Suppose P and Q are both true:
\(\displaystyle \left(\neg Q \wedge P \right) \not\equiv \left(Q \vee P \right)\)
\(\displaystyle \left(\neg \text{TRUE} \wedge \text{TRUE} \right) \stackrel{?}{\equiv} \left(\text{TRUE} \vee \text{TRUE} \right)\)
\(\displaystyle \left(\text{FALSE} \wedge \text{TRUE} \right) \stackrel{?}{\equiv} \text{TRUE}\)
\(\displaystyle \text{FALSE} \not\equiv \left(\text{TRUE} \right)\)
Instead, starting again from the last known good line, we can swap the order of P and Q and make the statement:
\(\displaystyle \left( \neg Q \vee P \right) \equiv \left(\neg Q \wedge P \right)\)
And from there, it should be obvious to see why the two expressions cannot possibly be equivalent.
