Question claims that conclusion (L v M) follows from given premises.
That is, [(P ^ Q ^ R) ^ [(Q <-> R) -> (L v M)]] -> (L v M) is a tautology.
So, we have to make conclusion False while Premises are True.
i.e [(P ^ Q ^ R) ^ [(Q <-> R) -> (L v M)]] = True
and (L v M) = False. So, lets do it.
To make conclusion (L v M) false. Our only option is L = False, and M = False.
Now, moving head to premises. Our goal is to make premises True.
We have to make (Q <-> R) false in order to make [(Q <-> R) -> (L v M)] true. For that L and M are false, so we make Q = true and R = false (or viceverse) in the premise [(Q <-> R) -> (L v M)] to make it true.
But, then the premis (P ^ Q ^ R) will assume over all value as false.
Now, let us substitute values of all propositional variables in given formulas.
L = False, M = False, Q= True, R= False and P = True. Lets take P = true (we can take P = false as well but it doesn't matter).
So, in [(P ^ Q ^ R) ^ [(Q <-> R) -> (L v M)]] -> (L v M) we have
[(T ^ T ^ F) ^ [(T <-> F) -> (F v F)]] --> (F v F)
= [(F) ^ [(F) -> (T)]] --> (F)
= [(F) ^ (T)] --> (F)
= [(F)] --> (F)
= T
So, we tried but fail to prove T --> F. Hence, we say (L v M) follows from [(P ^ Q ^ R) ^ [(Q <-> R) -> (L v M)]].
