Question

find the area of the shaded figure.

the area is
(simplify your answer.)

Explanation:

Step1: Identify the shape

The shaded figure appears to be a square? Wait, no, looking at the diagram, the total length of the side of the large square (if we consider the outer figure) would be \( y + 19 \), and the shaded area? Wait, no, actually, the shaded figure is composed of a square of side \( y \), a rectangle of \( y \times 19 \), a rectangle of \( 19 \times y \), and a square of \( 19 \times 19 \)? Wait, no, maybe it's a square with side \( y + 19 \), but the shaded area? Wait, no, the diagram shows that the shaded part is the sum of a square \( y \times y \), a rectangle \( y \times 19 \), a rectangle \( 19 \times y \), and a square \( 19 \times 19 \)? Wait, no, actually, the figure is a square with side length \( y + 19 \), so the area of the square is \( (y + 19)^2 \). Let's expand that.

Step2: Expand the square

Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \), where \( a = y \) and \( b = 19 \). So, \( (y + 19)^2 = y^2 + 2 \times y \times 19 + 19^2 = y^2 + 38y + 361 \). Wait, but maybe the shaded area is the area of the square with side \( y + 19 \). Let's check the diagram again. The figure has a square of side \( y \), a rectangle of \( y \times 19 \), a rectangle of \( 19 \times y \), and a square of \( 19 \times 19 \). So adding those areas: \( y^2 + 19y + 19y + 19^2 = y^2 + 38y + 361 \), which is the same as \( (y + 19)^2 \).

Answer:

\( y^2 + 38y + 361 \) (or \( (y + 19)^2 \))