Please explain the weird layout, i have no idea how to approach

It's pretty straightforward.

P(X) must sum to 1. You're given the first 4 values. Determine the 5th. The rest are 0.

\(E[X] = \sum~x p(x) = \sum \limits_{x=1}^4~x\cdot \dfrac{1}{x+2}+5k\)

Use the value of \(k=P(5)\) and evaluate that sum.

can u explain how k= 1/20 plz im still very confused, thanks

the probabilities must sum up to 1

\(1 = P[1]+P[2]+P[3]+P[4]+P[5] = \dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6} + k\\ \\ 1 = \dfrac{20+15+12+10}{60}+k\\ \\ 1 = \dfrac{57}{60}+k\\ \\ k=\dfrac{3}{60} = \dfrac{1}{20}\)