Find the largest interval for which the Existence and Uniqueness Theorem for Linear IVPs guarantees a unique solution for the following IVP.

\[ -{y'} e^{\left(-4 \, t\right)} = {y} t e^{t} + {\left(t + 4\right)} {\left(t - 5\right)} \hspace{2em} x( -4 )= -4 \]