Answer: B

Small Hint:

Adding $\triangle ABD$ to each of the two triangles produces two larger triangles

Big Hint:

Subtracting the shared area makes $[ADE] - [BDC]$ equal to $[ABE] - [ABC]$

Solution:

Let $[ABD]$ be the area shared by both large triangles. Then $[ABE] = [ADE] + [ABD]$ and $[ABC] = [BDC] + [ABD].$

Subtracting, $$\begin{aligned} &[ADE] - [BDC] \\ &= [ABE] - [ABC]. \end{aligned}$$ Since $\angle EAB$ and $\angle ABC$ are right angles, $$\begin{aligned} [ABE] &= \tfrac12(4)(8) = 16, \\ [ABC] &= \tfrac12(4)(6) = 12. \end{aligned}$$

The difference is $16 - 12 = 4.$

Thus, the correct answer is B.