The impulse response, $h(t)$, of a system I have this differential equation 
  $$
    y''(t) + y'(t) + y(t) = x(t) - 3x'(t)
    $$

I found the system response of the system: 

$$
        H(f) = \frac{1 - 6\pi fj}{(2\pi fj)^2 +2\pi fj+1}
    $$
which is equivalent to 

$$
        H(s) = \frac{1 - 3s}{s^2 +s+1}
$$

My question is: How to find the impulse response? If I factorize the denominator, it's not gonna be pretty. e.g., $s + \frac{1 +\sqrt{-3}}{2}$ 

Also, what does the following mean? 


Find the output for $t \ge 0$ with initial conditions $y(0) = 1$, $y'(0) = -1$ when the input is $2cos(6t)u(t)$. Identify the zero state, the zero input, the transient, and the steady state responses.
firstly do partial fraction for H(s) and then use transform tables of Laplace Transform. Using inverse laplace and tables you must be able to find impulse response. I think you can use j^2 instead of -1 and find complex roots for denominator. Then, using transform pairs you will find a complex solution.