The given ellipse is x^2/4 + y^2/1 = 1 i.e., the point A, the corner of the rectangle in 1st quadrant, is (2, 1). Again the ellipse circumscribing the rectangle passess through the point (4, 0), so its equation is
x^2/16+y^2/b^2=1
A(2,1) lies on the above ellipse.
4/16+1/b^2=1
1/b^2=1-1/4=3/4
b^2=4/3
Hence the equation of the desired ellipse is x^2/16+3/4 * y^2 = 1
x^2+12y^2=16