You took this second-order ODE apart into two first-order ODEs in one
of the previous homework problems:

1. get highest-order term by itself on LHS:

  x'' = t - xx'

2. define helper var:  

  v=x'

3. rewrite and stick on helper var eqn:

  v' = t - xv
  x' = v

The state vector has two variables in it: [x]
                                          [v]

At t=0, you are told, this vector has the following entries:

[x]    = [1]
[v]      [2]
   t=0

This means that you can calculate the entries of the derivative vector:

[x']    = [   2    ]
[v']      [0 - 2(1)]
   t=0

Here's the Adams-Bashforth (aka just "Adams") formula, from the
formula sheet:

   X(t_{n+1}) = X(t_n) + h/2 [3 F(X_n,t_n) - F(X_{n-1},t_{n-1})]

In this problem, X and F are two-vectors.  The only hard part now is
figuring out what to plug in where.  The first step is to recognize
that you need two points to fire up 2nd-order Adams-Bashforth, but
you're only given one (the vector [x] at t=0), so you need to estimate
                                  [v]
[x] at t=0.1 in order to use that formula.  One way to do that is with
[v]
forward Euler, or some other single-step method:

   [x]    = [1] + 0.1 [ 2] = [1.2]
   [v]      [2]       [-2]   [1.8]
     t=0.1

Now we have enough info to plug'n'chug.  In particular, t_{n-1}=0,
t_n=0.1, X_{n-1} = [1], and X_n = [1.2], so.....
                   [2]            [1.8]

X(0.2) = [1.2] + .05 { 3 [     1.8      ] - [   2    ] }
         [1.8]           [0.1 - 1.2(1.8)]   [0 - 1(2)]

       = [1.370]
         [1.591]  

Sorry the notation got so gross.