To solve this DE by Laplace Transform (LP) we need additivity and the two following properties of LP,
Given the DE
Take LP from both sides
Apply (1) and (2):
Use initial conditions: y(0)=-2, y'(0)=6 and group
What we did so far is the LP of the solution. To find the solution, y(t) we hat to take inverse LP from RHS of (3):
Check all details, there can be mistakes.