Explanation : Using Linearity of Expectation, we can write,
E[(X+2)2] = E[X2] + E[4X] + E[4]
The Poisson distribution, mean and variance are same. Here Mean is given as 5. So variance should also be 5.
Also,
Variance = E[X2] – (E[X])2
5 = E[X2] – 25.
E[X2] = 30
Thus E[(X+2)2] = 30 + 4*5 + 4 = 54.