Using superposition and time invariance to find RL circuit response
The answer to your last question is no. Nobody can explain how that answer was obtained because it's wrong. The fact that it is wrong is obvious, because due the term $e^{(t-2)}$ (with a positive power of $t$!), the output will keep increasing indefinitely, which is impossible for a stable system (such as the given RL-circuit).

Your answer is the correct one, and you must have made a mistake while checking the book's answer.