You are asked for two speeds.  But actually there are four speeds to
consider:

1.  The speed at which the plane would be traveling if there was no wind,
    (which is not the case).   Let's call this speed x

2.  The speed of the wind.  Let's call this speed w.

3.  The faster speed with the wind.  This is the actual faster speed of the
    plane when the wind is blowing in the direction the plane is flying.  This
    speed is x+w.

4.  The slower speed against the wind.  This is the actual slower speed of the
    plane when the wind is blowing in the opposite direction to the direction
    the plane is flying.  This speed is x+w.

So we use the formula rt = d substituting (x+w) for r, 2 for t, and 300 for d:

                      rt = d
                  (x+w)2 = 300

We divide both sides by 2:

                     x+w = 150

So we use the formula rt = d again but this time substituting (x-w) for r, 
2 again for t, but only 270 for d:

                      rt = d
                  (x-w)2 = 270

We divide both sides by 2:

                     x-w = 135

So we have this system of equations to solve:

                     x+w = 150
                     x-w = 135

Solve that system and get x = 142.5 mph and w = 7.5 mph

The wind speed is 7.5 mph and the speed of the plane in still air is 142.5 mph.

Checking:

With the wind the plane was traveling 142.5+7.5 or 150mph, and in 2 hours
he traveled twice 150 or 300 miles.  That checks.
 
And against the wind the plane was traveling 142.5-7.5 or 135mph, and in 
2 hours he traveled twice 135 or 270 miles.  That checks. So we know the 
answers are correct.

Edwin