You have constant coefficients. I would substitute instead.
Then you only need to solve .
x'' + 4x' -8x = 0
x(0) = 0 x'(0) = 1
So I set up the transform, and got it to (s^2)L(x) +4sL(x)-8L(x) = 1
From here, x(t) = L^(-1) of 1/(s^2+4s-8)
I did a sort of completing the square, to get 1/((s+2)^2-12), which almost fits the equation for a laplace transform of e^(at)cos(bt), but now this b value would be the square root of a negative number...where did I go wrong? Any clarification would be great, thanks!