Question

find the area of the shaded region.
2x + 8
x + 6
x (square with side x inside a triangle)

Explanation:

Step1: Find area of triangle

The triangle is a right triangle with base \( x + 6 \) and height \( 2x + 8 \). The formula for the area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). So, area of triangle \( = \frac{1}{2}(x + 6)(2x + 8) \).
Expand \( (x + 6)(2x + 8) = 2x^2 + 8x + 12x + 48 = 2x^2 + 20x + 48 \). Then, half of that is \( x^2 + 10x + 24 \).

Step2: Find area of square

The square has side length \( x \), so area of square \( = x \times x = x^2 \).

Step3: Subtract square area from triangle area

Shaded area = Area of triangle - Area of square. So, \( (x^2 + 10x + 24) - x^2 \). Simplify: \( x^2 - x^2 + 10x + 24 = 10x + 24 \).

Answer:

\( 10x + 24 \)