Laplace Transform Derivative (In context of differential equation)

Question

Solve

$$ \begin{equation*} \begin{cases} y''+5y'+6y=3e^{4t}\\ y(0)=1\\ y'(0)=-1. \end{cases} \end{equation*} $$

I solved it using Laplace Transforms.

However, the transform of derivatives only uses $y(0)$, and not $y(1)$ or $y(-2)$.

What if the IVP gave me $y(5)$ and $y'(5)$ instead? My solution would be in terms of $y(0)$.