Notes
four squares in a big square solution

Four Squares in a Big Square

Solution by Properties of Squares

The diagonal of the blue squares is half the side length of the outer square, which is $3$. Using the properties of a square, the area of a blue square is therefore $4.5$.

The sides of the blue square are at $45^\circ$ so form the diagonals of a square in the middle whose sides are the green lines and the sides of the red squares. This square is the same size as a red square, so the red squares have side length $2$ and area $4$.

The total shaded area is therefore $17$.