How to calculate the zero state response?
The answer to your question is essentially contained in Dilip's comment, but I guess as a beginner you need some more help. First, it's important to realize that the response of an LTI system to a scaled and shifted impulse
$$x[n]=a\delta(n-k)$$ is simply a scaled and shifted version of its impulse response:

$$y[n]=ah[n-k]$$

Furthermore, the response to a sum of signals equals the sum of the responses to the individual components of the input signal. Consequently, the response to the given input signal

$$x[n]=2\delta[n]-3\delta[n-2]$$

is

$$y[n]=2h[n]-3h[n-2]\tag{1}$$

Now you just have to express the impulse response $h[n]$ in terms of $\delta$-impulses. By noting that
$$u[n]-u[n-3]=\delta[n]+\delta[n-1]+\delta[n-2]$$ you get

$$h[n]=\delta[n]+\delta[n-1]+4\delta[n-2] \tag{2}$$

Plugging (2) into (1) gives the desired result.