If you know about polar coordinates (and especially equations of circles that are tangent to the origin) and you know about integration, then you could find these areas by placing the origin at $G$. You would make use of the radii ($3$ and $9$) to find the angles at which the $A$- and $B$-centered circles are oriented relative to your primary polar axis.
$$
\begin{align}
\text{red area} &= \text{area of }\Delta ABC + \text{area of semi-circle} - \text{area of quadrant} \\
&= \tfrac12 \times 21 \times 28 + \tfrac12 \pi \left(\tfrac{35}{2}\right)^2 - \tfrac14 \pi (21)^2 \\
&= 294 + 481.0563 - 346.3606 \\
&= 428.6957
\end{align}
$$
So it looks like you are right and your book is wrong.
Best Answer
You need to find the area of 4 remaining parts and subtract them from the area of the square.![enter image description here](https://i.stack.imgur.com/0BVHt.jpg)