Decide if L is regular or not and argue it. Trying to use Pumping Lemma
(a): Recall that regular languages are closed under complement, i.e if $L$ is regular, so is $L'$. ($F_{L'} = Q_L- F_L$)

Assume that $L$ is regular.
This follows: if $L = \{x \in \{0,1\}^* \mid |x|_0 \neq 4.|x|_1 \}$ is regular, then so is $L' = \{x \in \{0,1\}^* \mid |x|_0 = 4.|x|_1\} $

Now you should show that $L'$ is not regular. And that gives you a contradiction.

(b) You can construct the following DFA for this.