The normal expectation is that you provide some indication of what you have attempted so far - rather than hoping someone will supply the complete solution.
The Laplace method gives you a complete solution - both the transient and steady state components of the response are derived. You really only need the steady state responses.
You could apply some "simple' reasoning as follows
1. In the case of DC stimulus, the transfer function H(jw) reduces to a form in which w=0
So H(jw)=1/((1+jw)*(2+jw)) becomes H(0)=1/(1*2)=1/2 [A/V]. The output DC (current?) value is therefore 1/2 of the input DC voltage. Or 2A ..
2. If w=2 then H(j2) = 1/((1+j2)*(2+j2)) = -0.05-j0.15. Now think about what happens if a sinusoidal voltage 2cos(2t) is applied to the transfer function H(j2) - do it in the frequency [not Laplace] domain and then convert back to the time domain.
3. Split F(t) into the two components at w=2 and w=4 - you'll need to also determine H(j4). Determine the output response for each angular frequency. Phase shift due to the (linear) transfer function can be algebraically added to the initial phase values of the forcing function(s). Since the system is linear you can then add the two AC resulting components in 2w and 4w to give the overall response in the time domain.