let

x = the width of the frame

If the frame of the picture is 8 inches by 10 inches outside then the total area (together with the frame) will be

8 * 10 = 80 in^2

the area of the picture that shows is the area of a rectangle with ( 8 - x) and (10-x) sides

(8 - x) * (10 - x) = 48

80 - 8*x - 10*x + x^2 = 48

x^2 - 18*x + 32 = 0

the solutions of the quadratic equation are

x1 = 2 in - this is the solution of our problem

x2 = 16 in is not a solution since the width must be smaller then the sides ( 8 in and 10 in)

the width of the frame is 2 inches.