jmc

The large square, $ASHY$, is divided into seven regions. Two of these ($ABDC$ and $EFHG$) are squares. Four of the regions ($BSD,$ $CYD,$ $SFE,$ $YGE$) are right triangles. Finally, the seventh region is $DYES$, the quadrilateral whose area we wish to know. Thus, we subtract the area of the first six regions from the area of $ASHY$.

$ASHY$ has side length $5$ and therefore has area $5^2=25$.

The two small squares each have side length $1$ and thus area $1$.

The right triangles each have legs of length $1$ and $4$ (since, for example, $CY=AY-AC=5-1=4$). Thus, each right triangle has area $\frac{1\cdot 4}{2}=2$.

Subtracting the two small squares and the four right triangles from the large square, we determine that the area of $DYES$ is $$25-2\cdot (1)-4\cdot (2)=\boxed{15}.$$