Solution:

Firstly, let's label the points as follows:

Since we have a rectangle, $AB = 4.$ By the Pythagorean Theorem, we have $AC = 3.$ Then, since $\angle ACB$ and $\angle DCE$ are complementary, $\angle BAC = \angle CDE = 90^\circ,$ and $BC = CE$ we know $ABC \cong CDE.$ Therefore, $ED = 3$ and $EF = 1.$

Since $\angle CED$ and $\angle GEF$ are complementary and $\angle GEF = \angle CDE = 90^\circ,$ we know $EFG$ and $CDE$ are similar. This means $\dfrac{EC}{CD} = \dfrac{EG}{EF},$ so $EG = 1.25.$ Since the shaded region is a trapezoid, we can get the area as $$\dfrac{(GE+BC)EC}2 = \dfrac{ 5(5+1.25)}2$$$$= \dfrac{5\cdot 6.25}{2} = 15.625.$$

This is equal to $15 \dfrac 58.$

Thus, the answer is D .