To maximize the value of P(x,y), you must maximize the value of 4x + 3y.

At point B, P(x,y) = 4 • 0 + 3 • 4 = 12.

At point C, P(x,y) =4 • 4 + 3 • 0 = 16.

The value of P(x,y) is higher at point C then at point B and points B and C are connected by a straight line delineating an edge of the shaded region, no other points along this line or anywhere else in the shaded region can have a higher value than point C.