= Solution
The printed expression has one unmatched closing parenthesis. Remove that final surplus parenthesis and read the first disjunct as $\exists x\,P(x)$; the other disjunct has outer <universal quantification> over the displayed implication. To distinguish the two independent bound uses of $x$, rename the first to $u$.
Push the outer <negation> through the <logical disjunction> and through the <quantifiers>. Negating the implication preserves its existential antecedent and negates its consequent. Applying <De Morgan's laws> and the <quantifier> negation rules gives the <negation normal form>
$$
\boxed{\bigl(\forall u\,\neg P(u)\bigr)\land\exists x\left[\bigl(\exists y\,Q(x,y)\bigr)\land\bigl(\forall z\,\neg R(z)\bigr)\land\bigl(\exists t\,\neg T(x,t)\bigr)\right].}
$$
Each <negation> is now directly before an <atomic formula>. No <bound variable> has been captured, and no original <existential quantification> in the antecedent has accidentally been negated twice.
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