I'm trying to "Solve y'' - 3y' + 4y = 0; y(0) = 1, y'(0) = 5" (using Laplace transforms) and I realize my first mistake was where I put the red in the first attached document.

(The second attached document is the continuation of my incorrect work just for the sake of completeness).

How do I get to the final answer

y(x) = e^[(3/2) x] cos(sqrt(7)/2 x) + sqrt(7) e^[(3/2) x] sin(sqrt(7)/2 x)

from my last correct step?

(My book doesn't show the steps to get from where I'm stuck to the final answer).

Any help would be greatly appreciated!

Thanks in advance!